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Tropical varieties, maps and gossip

Citation for published version (APA):Frenk, B. J. (2013). Tropical varieties, maps and gossip. Eindhoven: Technische Universiteit Eindhoven.https://doi.org/10.6100/IR750815

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TROPICAL VARIETIES, MAPS AND GOSSIP

Bart Frenk

This work is licensed under a Creative Commons Attribution

3.0 Unported License, which can be found via:

http://creativecommons.org/licenses/by/3.0/

Tropical varieties, maps and gossip / B.J. Frenk

Technische Universiteit Eindhoven 2013

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-3343-5

NUR: 918

Cover design: Madelief Brandsma

Printed by: Koninklijke Wöhrmann, Zutphen

Printed on: 90g chlorine-free Biotop paper

Tropical varieties, maps and gossip

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen

op woensdag 13 maart 2013 om 16.00 uur

door

Bartholomeus Johannes Frenk

geboren te Schiedam

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A.M. Cohen

Copromotor:dr.ir. J. Draisma

Preface

Tropical mathematics, or tropical geometry are loosely defined terms referring to a

subfield of mathematics in which a central role is played by an algebraic structure

called the tropical real numbers. This is the set RYt8u equipped with the operations

of addition and taking the minimum. This structure first found applications in the

algebraic formulation of optimization problems (cf. [16]). In tropical mathematics,

however, the emphasis is on geometry over the tropical semifield. For this thesis it is

convenient to make a very rough division of tropical mathematics into two (overlap-

ping) parts.

One part studies images of embedded algebraic varieties under a valuation. Such

images, called tropicalizations or tropical varieties, are (underlying sets of) polyhedral

complexes that have a characterization in terms of the algebra of the tropical real

numbers and the ideal of the variety. The founding idea is that the properties of

tropicalizations reflect properties of the algebraic variety in question. One can then

use combinatorial techniques to derive algebraic geometric results, and vice versa.

A celebrated example is G. Mikhalkin’s computation of Gromov-Witten invariants by

counting tropical curves in [45].The other part studies the algebra and geometry of the tropical real numbers with-

out reference to a valued field. For example, it includes functional analysis over the

tropical real numbers (cf. [43]), and studies abstractions of tropical varieties. A rea-

son to study such abstractions is that they are applicable to the aforementioned part

of tropical geometry, while the techniques involved do not depend on the characteris-

tic of the field. As argued in Chapter 4 the concept of a matroid falls naturally within

this part of tropical geometry.

Chapters 1 and 2 fit within the first part. The first chapter deals with reformulating

and proving the main properties of the tropicalization of an algebraic variety. None of

these results are new, but as far as we know have not been collected in a published

source. The second chapter explores the relation between unirationality of algebraic

varieties and tropical unirationality of its tropicalization.

Chapter 3, 4 and 5 fit within the second part. Chapter 4 is the largest chapter of

this thesis. Its aim is to construct a category whose objects are tropical linear spaces.

i

ii

These spaces are abstractions of tropicalizations of vector spaces and are closely re-

lated to the valuated matroids of [23]. The reason for constructing such a category

is that in general maps between tropical varieties are problematic. For example, in

the context of tropical linear spaces the naive notion of a linear map does not map

subspaces to subspaces. Chapter 4 attempts to remedy that situation.

Chapter 3 puts the tropical semifield into the broader context of idempotent semi-

fields. Its main purpose is to understand which algebraic properties of idempotent

semifields are essential for Chapter 4. The last chapter, Chapter 5 is about the monoid

generated by distance matrices under tropical matrix multiplication. This monoid is

related to the gossip problem and referred to as the gossip monoid. We make use of

techniques related to realizing a finite metric by a graph to compute polyhedral fan

structures on the gossip monoids of square matrices of size at most 4.

Acknowledgements

To me, sometimes an appropriate metaphor for being a Ph.D. candidate in mathe-

matics is tropical. It is that of deep, dark jungle. An environment without clear paths,

in which it is often difficult to distinguish the friendly from the fearful, at least at

first sight. A place easy to get lost in. That I came out in one piece is greatly due

to the efforts of my guide and copromotor, dr. ir. Jan Draisma. Of the many things

I learned from him the most important was probably to face unknown mathematics

with courage, no matter how fearsome it might appear. He also wrote the original

Tropical Algebraic Groups NWO project proposal that was the starting point of this

thesis.

During exploration one meets the people along the way that helped the project

progress in some way or another. They appear here in no particular order.

I would like to thank Prof. Dr. Eva Maria Feichtner for the invitation to do part of

my thesis work at the University of Bremen, where I spent three months. Life there

was made easier, both professionally and personally, by the people of the DiscreteStructures in Algebraic Geometry group and the capoeristas of Ginga Brasil Bremen. I

extend my thanks.

My promotor, prof. dr. Arjeh Cohen, was kind enough to accept me as his official

Ph.D. student. The three aforementioned people were also part of my reading com-

mittee, along with Prof. Dr. Thorsten Theobald and Prof. dr. Andries Brouwer, all of

which I would like to thank for their helpful corrections and comments on the first

draft of my thesis. It goes without saying that any errors and omissions left in the

thesis are entirely mine. The reading committee members are also part of my defence

committee, along with Prof. Dr. Michael Joswig en Prof. dr. Peter Butkovic. I would

like to thank all of them for their time and effort.

iii

My colleagues and fellow Ph.D. candidates at the Eindhoven University of Technol-

ogy. In particular, Jan-Willem Knopper, for his patience in dealing with the computer-

related problems I sometimes had, and Maxim Hendriks, that guided me gently

through the myriad of choices one has to make when deciding how to print one’s

thesis.

Of course, not all of my time in the last four years was spent exploring this

metaphorical jungle.

The capoeira-lessons of Mestre Tayson in Eindhoven were always a very good way

to take my mind of the mathematics. I would like to thank him and his students, my

friends, in no particular order: Danni, Damian, Tsveti, Mike, Roelof, Dirk, Imre, Qorin,

Matilde.

I would like to thank my parents for their continuous support. I consider myself

lucky that my twin sister Myrthe worked on a Ph.D. project at the University of Maas-

tricht while I worked on mine in Eindhoven. This was particularly helpful during the

periods of frustration that are bound to come up when doing research. Talking to her

about such matters was always helpful for clearing my mind. Finally, great thanks

goes to my girlfriend, Hilde, who was instrumental in the entire process for her un-

wavering support.

Contents

Preface i

List of notations viii

1 Tropical geometry 1

1.1 Quick algebraic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Tropical geometry of hypersurfaces . . . . . . . . . . . . . . . . . . . . . . 9

1.4 Tropical geometry in higher codimension . . . . . . . . . . . . . . . . . . 11

1.5 Tropical geometry and valuation theory . . . . . . . . . . . . . . . . . . . 18

2 Tropical unirational varieties 21

2.1 Some classes of tropically unirational varieties . . . . . . . . . . . . . . . 23

2.2 Combining reparameterizations . . . . . . . . . . . . . . . . . . . . . . . . 26

2.3 Birational projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Very local reparameterizations . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Algebra of idempotent semirings 37

3.1 Modules over semirings and homomorphisms . . . . . . . . . . . . . . . . 37

3.2 Linear functionals on idempotent modules . . . . . . . . . . . . . . . . . . 48

3.3 Ranks and vanishing conditions . . . . . . . . . . . . . . . . . . . . . . . . 52

3.4 Modules over linearly ordered semirings . . . . . . . . . . . . . . . . . . . 58

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 A category of tropical linear spaces 61

4.1 Valuated matroids and tropical linear spaces . . . . . . . . . . . . . . . . 63

4.1.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.1.2 Basic constructions and relations . . . . . . . . . . . . . . . . . . . 71

4.1.3 Parameter spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Functions and their graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

v

vi CONTENTS

4.2.1 Linear functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.2 Extensions of matroids . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3 Morphisms between linear spaces . . . . . . . . . . . . . . . . . . . . . . . 113

4.3.1 A category of tropical linear spaces . . . . . . . . . . . . . . . . . . 113

4.3.2 The tropical linear monoid . . . . . . . . . . . . . . . . . . . . . . . 121

5 The gossip monoid 127

5.1 Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.2 Relation to the gossip problem . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3 Polyhedral structure of the gossip monoid . . . . . . . . . . . . . . . . . . 134

5.3.1 Graphs with detours . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.4 Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.4.1 n¤ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

5.4.2 n� 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Bibliography 145

Index 150

Summary 153

Curriculum Vitae 155

List of notations

pv : wq dual residuum of v by w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

rns the numbers t1, . . . , nu

α� transpose of the map α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

µ the independent set valuated matroid of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

F extended semifield of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

B the trivial semifield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

BpX q Berkovich analytic space of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Cttεuu the field of Puiseux series over C in ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

DrdpL q parameter space of d-dimensional tropical linear subspaces of L . . . . . . . . 80

ExtppL q elemtary extensions of the tropical linear space L . . . . . . . . . . . . . . . . . . . . . . 97

Extppµq elementary extensions of the valuated matroid µ . . . . . . . . . . . . . . . . . . . . . . . . 97

GrdpKnq the Grassmannian of d-dimensional linear subspaces of Kn (in its Plücker

embedding)

I pX q radical ideal of polynomials vanishing on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

εR indicator of the semiring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

invpgq initial form of the polynomial g with respect to the weight vector v . . . . . . . . 5

xv, wy the tropical inner product of v and w . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

λrX , Y s restriction of the linking system λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

λΓ linking system induced by Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

¤� natural order on an idempotent commutative semigroup . . . . . . . . . . . . . . . . . 40

vii

viii CONTENTS

λK the cofactor linking system of λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .106

Lµ tropical linear space of the valuated matroid µ . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

matpµq the underlying matroid of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

µ{X contraction of X , i.e. µ � RzX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

µ � X contraction of µ to X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

µY ν (valuated) matroid union . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

µ` ν (valuated) matroid direct sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

µrX s restriction of X of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76

M op opposite module of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

φ^ dual residual of the increasing map φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

respµq the minimal, or residual, matroid of µ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

TGLn the tropical linear monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

Troppgq tropicalization of the polynomial g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

T pSq the tropical prevariety of the set of polymials S . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

R� group of units of the semiring R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

DnpFq distance matrices with entries in F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Ei jpaq elementary distance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Frxs the semiring of polynomials over F in x� tx1, . . . , xnu. . . . . . . . . . . . . . . . . . . . . 2

Jλ the matrix all of whose entries are λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

M� algebraic dual of the module (over a semiring) M . . . . . . . . . . . . . . . . . . . . . . . . 48

SnpFq the gossip monoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

vK set of all w such that xv, wy vanishes tropically . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

vI generator of a tropical linear space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .68

Q8 the tropical rational numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

R8 the tropical real numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Z8 the tropical integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2

Chapter 1

Tropical geometry

The tropicalization map is the main tool in tropical geometry. Roughly, it assigns a

polyhedral complex to an embedded algebraic variety. This chapter collects the foun-

dational results from the tropical geometry literature on the tropicalization map. It

focuses on three theorems: Bieri-Groves, the Fundamental Theorem of Tropical Ge-

ometry, and the Finite Tropical Basis Theorem, which we refer to as the main theo-rems of tropical geometry. They are respectively Proposition 1.4.9, Theorem 1.4.8 and

Corollary 1.4.7.

The route to proving the three main theorems of tropical geometry is through the

device of generic projections that was first applied in the paper by Bieri and Groves

[7]. The advantage are twofold: (i) it allows one to prove both the Fundamental

Theorem and the Bieri-Groves theorem by reducing to the hypersurface case, (ii) the

existence of a finite tropical bases is an immediate consequence of the main lemma

of the method. The proofs we give are not new. They are based on those in [52], [7]and [32].

1.1 Quick algebraic definitions

A semifield is an algebraic structure with a multiplication and an addition operator

that satisfies all the axioms of a field save the one that states that addition is invertible.

In that sense they are fields without a minus. In this thesis, and especially the first two

chapters, the semifields under consideration are of two types: (algebraically closed)

fields, or tropical semifields, which we define next.

The tropical real number semifield is the semifield R8 � pRY t8u,min,�,8, 0q.

In this semifield the "addition" operation is given by min, the "multiplication" is given

by �. It is common to write x ` y for mintx , yu and x d y for x � y , to stress

the roles of both operators. We will usually follow this convention. The tropical real

1

2 CHAPTER 1. TROPICAL GEOMETRY

numbers do not form a field, since for any a P R there is no x P RY t8u such that

a ` x � minta, xu � 8. Notice that 8 is the neutral element for tropical addition,

and 0 is the neutral element for tropical multiplication.

A tropical semifield is a subsemifield of R8. Their underlying sets are of the form

G Y t8u where G is a subgroup of pR,�, 0q. The tropical semifield with underlying

set G is denoted G8. For example, Q8 is the semifield of tropical rational numbers

and Z8 is the semifield of tropical integers. We reserve B for the tropical semifield

t0,8u and refer to it as the trivial semifield.

1.2 Foundations

Let F be a semifield and x � tx1, . . . , xnu a finite set of formal variables. A monomialin x is an element of the free commutative monoid generated by the elements of x. A

polynomial is an element of the free F -algebra generated by this monoid (having only

a finite number of coefficients different from 0 P F). The collection of all polynomials

in x over F forms an F -algebra itself and is denoted Frx1, . . . , xns, or simply Frxs.

It is common to think of polynomials as functions F n Ñ F . Let f P Frxs and

v P F n. The evaluation of f in v is the element of F obtained by substituting all

occurrences of x i in f by vi and replacing the formal operations by the operations

in the semifield. We denote it f pvq. The association of v ÞÑ f pvq to f defines a map

Frxs Ñ tF n Ñ Fu. When F is an infinite field this map is injective. However, it is

not for general semifields, even when F is infinite. In particular, when F is a tropical

semifield this map is not injective.

Example. The simplest example of the non-injectivity of the evaluation map is when

F � B and n� 1. The non-injectivity of the map is clear since the number of functions

BÑ B equals 4, while the number of polynomials in Brxs is not even finite. Moreover,

a2 � a for any a P B.

A more interesting example is the tropical univariate polynomial x2 `p1d xq` 2

over R8. It induces the function R8Ñ R8 given by v ÞÑmint2v, v�1, 2u in ordinary

notation. It is thus the minimum of three linear functions, the graphs of which are

depicted in Figure 1.1. It induces the same function as x2 ` pcd xq ` 2, when c ¥ 1.

For the remainder of this section F is a tropical semifield and K is a field equipped

with a surjective valuation ω : K Ñ F . We write R for the valuation ring ta P K |

ωpaq ¥ 0u and m for its maximal ideal ta P R | ωpaq ¡ 0u. The residue field R{m is

denoted k.

Example. A good example of such a field to keep in mind is that of the Puiseux seriesover C in ε, denoted Cttεuu. It is the union of all formal Laurent series fields Cppε1{nqq

over C in ε1{n, where n ranges over the positive integers. The order of such a formal

1.2. FOUNDATIONS 3

2

x2

1� x

Figure 1.1: The graph of the function induced by x2 ` p1d xq ` 2.

Laurent series is the minimal exponent occurring with a non-zero coefficient. For

example,

4ε�10 � ε1{2 � pi� 1qε6{8 � 20ε50,

is an element of Cppε1{2qq of order �10. The order of the trivial series 0 is defined to

be 8. The order maps on the different sets Cppε1{2qq of formal Laurent series coincide

on the intersections and hence extend to an order map on the union Cttεuu. This

extension, denoted ord, is a surjective valuation Cttεuu Ñ Q8. The field of Puiseux

series is algebraically closed (cf. [54]) and has residue field C.

Let f P Frxs and write,

f �àαPApcαd xαq,

where A is a finite subset of Nn and cα P Fzt8u. The tropical notation serves to

remind the reader of the nature of the polynomial. The tropical vanishing locus of fis the collection of points v P F n for which either the minimum in f pvq is attained for

two distinct α, or for which f pvq � 8.

Let g P Krxs. The tropicalization of g is the polynomial with coefficients in Fobtained from g by replacing the coefficients with their image under ω. With slight

abuse of notation it is denoted Troppgq and g ÞÑ Troppgq defines a surjective map

from polynomials over K to polynomials over F ,

Trop : Krx1, . . . , xns Ñ Frx1, . . . , xns , Tropp¸αPA

cαxαq �àαPApωpcαq d xαq. (1.1)

The tropical prevariety!of a polynomial associated to g is, by definition, the tropical

vanishing locus of Troppgq, as defined in the previous paragraph. It is denoted Tωpgq,or T pgq, when reference to ω is clear from the context. Since the map (1.1) is surjec-

tive the set of tropical vanishing loci of polynomials over F equals the set of tropical

prevarieties of polynomials over K .


L1

L2

L3

x

y

Figure 1.2: The tropical prevariety of the polynomial f � pε2 � ε3qx � y � ε�1

Remark. When F � R8 the function induced by a non-zero tropical polynomial frestricts to a piecewise linear function Rn Ñ R. The vanishing locus of the polynomial

is the union of the topological closure in Rn8 of the corner locus of this restriction and

the collection of points on which f attains the value 8. In particular, it depends only

on the function induced by f .

In fact the case F � R8 is the only case we need to consider. The reason is that

any tropical semifield F embeds in R8, and hence any polynomial f over F is also

a polynomial over R8. It is an immediate consequence of the definition that the

vanishing locus of f in F n equals the vanishing locus of f as a polynomial over R8intersected with F n.

Example. Let F � Q8 and K the field Cttεuu with the order valuation ord : K Ñ F .

Consider the polynomial

f � pε2 � ε3qx � y � ε�1.

in the variables x and y . Its algebraic vanishing locus is a line in the affine plane over

Cttεuu. Moreover, Tropp f q is the tropical polynomial,

Tropp f q � p2d xq ` y ` p�1q,

whose vanishing locus is the union of three half-lines L1, L2 and L3, given by the

equalities and inequalities (specified in terms of the classical operations on Q8),

L1 : 2� x � y ¤ �1,

L2 : 2� x � �1 ¤ y,

L3 : y � �1 ¤ 2� x .

It is depicted in Figure 1.2.

Notation and terminology extend to arbitrary sets of polynomials. Let S �

Krx1, . . . , xns. The tropical prevariety associated to S is the intersection of the tropi-

1.2. FOUNDATIONS 5

cal prevarieties of the elements of S and denoted T pSq, or TωpSq when we need to

specify the valuation, i.e.,

TωpSq �£gPS

Tωpgq.

A tropical basis of S is a subset B � S such that the intersection of T pgq, with g P B,

equals T pSq. A tropical basis always exists (take B � S), but it is not a priori clear

whether a finite subset always suffices.

Tropical prevarieties are general objects and, at least in this chapter and the next,

we restrict ourselves to tropical prevarieties of ideals in Krx1, . . . , xns. Such preva-

rieties are referred to as tropical varieties over K , or over pK ,ωq, if we need to be

precise. A prevariety is simply said to be a tropical variety if it is a tropical variety

over some pair pK ,ωq. Given F � ωpKq, the distinction is irrelevant when only con-

sidering principal ideals, but to prove this we require a rather technical lemma and

some definitions.

We henceforth assume that there exists a good section of ω, i.e., a map s : F Ñ Kthat assigns to each v P F an element spvq P K such that ωpspvqq � v, spv d wq �spvqspwq andωpspvq�spwqq � v`w. For example, in the case of the Puiseux series in

ε over C a good section is spvq ÞÑ εv . The choice of a specific section is non-canonical

and not strictly necessary, but makes the proofs more intuitive, at the expense of

making them less general. We fix such a section for the remainder of this chapter and

use the suggestive notation t v for the image of v under this section. Note that t8 � 0.

In [52] the choice of a section is avoided by making use of so-called twisted coordinatesubrings of Krxs.

Let g P Krxs be a polynomial and v P F n. Let w � Troppgqpvq and suppose that

w � 8. The initial form of g with respect to v is the image of t�w gpt v1 x1, . . . , t vn xnq

under the canonical projection Rrxs Ñ krxs. It is denoted invpgq.

Example. Let f � pε2�ε3qx� y�ε�1, as in the previous example. Then, spvq � t v �

εv is a good section of ord, and

inp0,1qp f q � 1,

inp�3,0qp f q � x � 1.

Parts (ii) and (iii) of the next (technical) lemma state important properties of ini-

tial forms. They are the main technical devices for proving that tropical prevarieties of

polynomials are tropical varieties and that the tropicalization of an algebraic variety,

to be defined later, doesn’t depend on a defining ideal. Part (i) essentially states that

the Gauss norm restricted to Krxs is multiplicative (cf. [10]).


Lemma 1.2.1 The following statements hold.(i) The map that assigns to each g P Krxs the function F n Ñ F defined by v ÞÑ

Troppgqpvq is multiplicative.(ii) Let v P F n. Denote the collection of polynomials g P Krxs that have Troppgqpvq �

8 by Dv . Then inv : Dv Ñ krxs is multiplicative.(iii) Let v P F n. Then v P T pgq if and only if either invpgq is not a monomial, or

Troppgqpvq � 8.

Proof. We prove the claims in the order stated.

(i) Let f , g P Krxs and v P F n. We need to prove that the equality Tropp f gqpvq �Tropp f qpvq d Troppgqpvq holds. Assume without loss of generality that v � p0, . . . , 0q

and that Tropp f qpvq � Troppgqpvq � 0. Then f , g P Rrxszmrxs. It is now sufficient to

show that f g P Rrxszmrxs. Since mrxs � mRrxs and the projection Rrxs Ñ Rrxs{mrxsis an R-algebra homomorphism it is suffices to prove that Rrxs{mrxs is an integral

domain. This is clear, since it equals krxs.(ii) Denote the map Kn Ñ Kn defined by pa1, . . . , anq ÞÑ pt v1 a1, . . . , t vn anq by σ

and write σ� for its pull back Krxs Ñ Krxs. Let f , g P Dv . Set w � Tropp f gqpvq,w f � Tropp f qpvq and wg � Troppgqpvq. Then w � w f � wg , by part (i). Denote the

canonical projection Rrxs Ñ krxs by π. Consider,

invp f gq � πpt�wσ�p f gqq

� πpt�wσ�p f qσ�pgqq

� πpt�w fσ�p f qt�wgσ�pgqq

� πpt�w fσ�p f qqπpt�wgσ�pgqq

� invp f q invpgq.

This proves the second claim.

(iii) Suppose v P T pgq and write g �°αPA cαxα. Let β P A be the index of

a term at which the minimum in Troppgqpvq is attained. The coefficient of xβ in

gpt v1 x1, . . . , t vn xnq equals cβ t v�β , where v � β is the ordinary inner product on Rn

extended to Rn8. The valuation of this coefficient equals Trop pgqpvq and hence the

coefficient of πpxqβ if invpgq is non-zero. Vice versa, if the coefficient of some πpxqβ

is non-zero in the initial form, then the corresponding term of g attains the minimum

in Trop pgqpvq. This proves the statement. l

Proposition 1.2.2 Let f P Krxs and write I for the ideal generated by f . Then Tωp f q �TωpIq. In particular, tropical prevarieties of polynomials are tropical varieties.

Proof. Since f P I it holds that TωpIq � Tωp f q. Let v P Tωp f q and g P I . There

exists h P Krxs such that g � hf . If either Tropp f qpvq � 8, or Tropphqpvq � 8,

1.2. FOUNDATIONS 7

then Troppgqpvq � Tropphqpvq d Tropp f qpvq � 8, by part (i) of Lemma 1.2.1. If

Tropp f qpvq � 8 and Tropphqpvq � 8 then invpgq � invphq invp f q. Since invp f q is

not a monomial invpgq is also not a monomial and hence g vanishes tropically on v.

This shows that v P T pIq. l

The next statement shows that the tropical variety of an ideal only depends on

the algebraic variety defined by this ideal. It is the essential ingredient in defining the

tropicalization of an algebraic variety, which do we after we prove the lemma.

Proposition 1.2.3 Let I � Krxs be an ideal and denote its radical by rpIq. ThenTωpIq � TωprpIqq.

Proof. Clearly, T prpIqq � T pIq, since rpIq � I . Let v P T pIq and f P rpIq. If

Tropp f nqpvq � 8 then Tropp f qpvq � 8, by Lemma 1.2.1, and hence v P T p f nq. Sup-

pose that Tropp f nqpvq � 8. Then invp f nq � invp f qn. Since invp f nq is not a monomial,

invp f q isn’t. Thus v P T f . This proves that T prpIqq � T pIq. l

Let X � Kn be an algebraic variety and let I � Krxs be an ideal whose algebraic

variety is X . Define the tropicalization of X as T pIq. By Proposition 1.2.3 it does not

depend on the specific choice of ideal. We denote it by Trop pX q. A tropical basis of

Trop pX q is defined to be a tropical basis of I pX q, i.e. a set of polynomials that vanish

on X whose tropical variety equals Trop pX q. Here we write I pX q for the radical ideal

of all polynomials vanishing on X . Note that we do not require that a tropical basis

generates I pX q, or even cut out the variety X .

Proposition 1.2.4 Let X � Kn be an algebraic variety such that X � Y Y Z, withY, Z � Kn algebraic varieties. Then Trop pX q � Trop pY q Y Trop pZq.

Proof. Let I � I pY q and J � I pZq. Then V pI Jq � X . Thus, Trop pX q � T pI Jq.Since I J � I , J it holds that Trop pX q � Trop pY q Y Trop pZq. Let v P Trop pX q and

suppose that v R Trop pY q. There exists f P I such that v is not in T p f q. By Lemma

1.2.1 the initial form invp f q exists and is a monomial. Let g P J and suppose that

Troppgqpvq � 8. Then invp f gq � invp f q invpgq is not a monomial, and hence invpgqis not a monomial, a contradiction. Thus, v P Trop pZq. l

Example. By definition every tropical variety is a tropical prevariety, but the con-

verse does not hold. As a simple example, let F � R8 and consider the tropical

prevariety T associated to the set of polynomials tx1 � x2 � 1, x1 � x2u. It is the

intersection of the tropical vanishing loci of x1 ` x2 ` 0 and x1 ` x2, which equals

tpv, vq | v P F and v ¤ 0u. Suppose that there exists an ideal I � Krx1, x2s such that


TωpIq � T . Then I is not principal. Suppose it were and set pgq � I . Write,

g �d

i�0

mi xi2, mi P Krx1s and d ¡ 0.

Let w1 P F with w1 ¡ 0 and consider Troppgqpw1, x2q P Krx2s. This polynomial has

positive degree and hence there exists w2 P F such that w2 is in the vanishing locus

of Troppgqpw1, x2q. Thus, pw1, w2q P T pIq and pw1, w2q R T .

If I is not principal its associated variety is either empty or a finite set of points.

Since the tropical variety does not depend on the choice of ideal, T pIq � T pI pX qq.In the first case 1 P I pX q, by Hilbert’s Nullstellensatz, and hence T pIq � H. In the

second case, X is a finite union of points and hence Trop pI pX qq is a union of tropical

varieties of ideals of the form px1 � p1, x2 � p2q. In particular, it is a finite union of

points. This shows that T is not a tropical variety.

We finish the section with a lemma, which serves as the base from which to prove

the main theorem of tropical geometry for hypersurfaces. In fact, one could say the

lemma is the main theorem of tropical geometry for points on a line.

Lemma 1.2.5 (Newton-Puiseux) Let f P Krxs be a univariate polynomial. ThenωpV p f qq � T p f q.

Proof. Since K is algebraically closed there exists a P K and b1, . . . , bd P K such that

f � apx � b1q . . . px � bdq. Let c P V p f q. Then c � bi , for some i � 1, . . . , d. If bi � 0

then by Lemma 1.2.1 the image of ωp0q under Tropp f q is 8 and hence ωp0q P T p f q.If bi � 0, then inωpcqp f q is a product of d initial forms inωpcqpx � b jq, at least one of

which is not a monomial. Hence inωpcqp f q is not a monomial. Thus, ωpcq P T p f q.

Reversing the above proof yields that T p f q �ωpV p f qq. l

Remark. The above lemma implies in particular that tropical semifields that appear

as the range of a surjective valuation on some algebraically closed field satisfy that

any non-constant polynomial over them has a non-empty vanishing locus. Such a

semifield F is divisible, i.e. it has the property that for all a P F the equation xn � a has

a solution. Over such semifields it holds that any polynomial function is the product

of linear polynomial functions. The emphasis here is on the word function, since it

does not necessarily hold on the level of polynomials.

As an example, consider the polynomial x2 ` p2d xq ` 1. Suppose that it decom-

poses as px ` aq d px ` bq � x2 ` ppa` bq d xq ` pad bq. Then, by a` b � 2, either

a � 2 or b � 2. Assume without loss of generality a � 2 and hence b � �1. But then

a ` b � �1 � 2. However, x2 ` p1d xq ` 1 and x2 ` p2d xq ` 1 define the same

function, and px ` 1q d px ` 1q � x2 ` p1d xq ` 1.

1.3. TROPICAL GEOMETRY OF HYPERSURFACES 9

It is interesting to note, however, that there is a multiplicative section of the map

f ÞÑ pv ÞÑ f pvqq, assigning the induced function to a polynomial. This section assigns

to each polynomial function the polynomial with minimal coefficients. See [30] for

details.

We can now state the theorems we aim to prove in this chapter. The first, usu-

ally referred to as the Fundamental Theorem of Tropical Geometry, is the state-

ment that the tropicalization of an algebraic variety X � Kn equals ωpX q, where

ωpp1, . . . , pnq � pωpp1q, . . . ,ωppnqq, with slight abuse of notation. The second state-

ment, referred to as the Bieri-Groves theorem is the statement that the tropicalization

of X is the topological closure of (the underlying set of) a polyhedral complex in Rn,

when F � R8. The third statement is that any tropical variety has a finite tropical

basis.

The next section is dedicated to proving these three statement for the case where

X is a hypersurface.

1.3 Tropical geometry of hypersurfaces

This section deals with tropical varieties of algebraic hypersurfaces in Kn. The reason

why the distinction between hypersurfaces and general algebraic varieties is made is

essentially Proposition 1.2.2. Every tropical variety of a hypersurface is the vanish-

ing locus of a tropical polynomial and vice versa. In particular, given F there is no

dependence on the field, as long as it is algebraically closed.

We begin by stating and proving the main theorem of tropical geometry for the

case of hypersurfaces. This specific case is also referred to as Kapranov’s theorem and

it appears, with a different proof, in [24]. The proof here is essentially that in [52].For each n P N the valuation ω : K Ñ F induces a map Kn Ñ F n, given by

ωpp1, . . . , pnq � pωpp1q, . . . ,ωppnqq, that we also denote by ω. The image ωpX q is

referred to as (non-Archimedean) amoeba of X in [24].

Theorem 1.3.1 (Kapranov) Let X � Kn be an irreducible hypersurface. ThenTrop pX q � ωpX q. Moreover, for each v P Trop pX q the set ω�1pvq X X is Zariski densein X .

Proof. Let v P Trop pX q. We can assume without loss of generality that v � p0, . . . , 0q.

Let I � Krxs be the ideal I pX q of X . There exists f P Rrxszmrxs with I � p f q and by

Lemma 1.2.2 it holds that T p f q � T pIq � Trop pX q. The tropical polynomial Tropp f qvanishes at v and hence, either (i) Tropp f qpvq � 8, or, (ii) the projection f of f in

krxs � Rrxs{mrxs has at least two terms.

In the first case, let p be a point in Kn with ωppq � v. Then f ppq � 0, since

ωp f ppqq ¥ Tropp f qpvq � 8, and hence p P X .


In the second case, there exists a variable x l which occurs with at least two distinct

exponents d and e in f . Suppose without loss of generality that l � n and d e. Write,

f � f0 � f1 xn � . . .� fd xdn � . . .� fe x e

n � . . .� fd xdn , fi P Krx1, . . . , xn�1s

Let p P pRzmqn�1 such that fdppq � 0 and feppq � 0. Such p exists since K , and hence

k is algebraically closed. Set gpyq � f pp, yq. The projection grys P krys has at least

two non-zero terms, and hence a non-zero root pn. Let pn P Rzm such that it projects

onto pn. Then, ωpgppnqq ¡ 0 and hence ωppnq P T pgq. By Lemma 1.2.5 there exists

pn P V pgq such that ωppnq � 0. Thus, pp, pnq P V p f q and ωpp, pnq � p0, . . . , 0q.

The projection of X Xω�1p0, . . . , 0q onto the first n � 1 coordinates contains a

Zariski dense set in Kn�1. Hence, X Xω�1p0, . . . , 0q it itself dense in X . l

It remains to establish the polyhedral structure of Trop pX q. Let f P Frxs and write,

f �àαPApcαd xαq, A� Nn and cα P Fzt8u.

The Newton polytope of f is the convex hull of A. The subdivision of the Newton

polytope induced by f is the regular subdivision of the Newton polytope associated to

the map that assigns the coefficient of xα to α P Nn. More precisely, it is the projection

of the lower convex hull of the set tpα, cαq | α P Au � Rn �R to Rn.

Theorem 1.3.2 Let X � Kn be the hypersurface defined by a polynomial g P Krxs. LetF � R8. The tropical variety Trop pX q is the underlying set of a pure polyhedral complexof codimension one that is dual to the subdivision of the Newton polytope associated toTroppgq.

Proof. Let f � Troppgq and write f �ÀαPA mα as a sum of monomial terms mα, i.e.,

mα is the formal product of a coefficient in F and a monomial. The function F n Ñ Fdefined by mα is an affine linear function on F , and hence v ÞÑ f pvq is the minimum

over a finite number of affine linear functions. A point v P F n lies on T pgq if and only

if the minimum is attained in at least two such affine linear functions. Thus, T pgq is

the union of the sets Cα,β � tv P F n | mαpvq � mβpvq ¤ mγpvq for all γ P Aztα,βuu,

some of which may be empty. The non-empty Cα,β ’s have dimension n� 1 and form

the maximal cells of a polyhedral complex. Moreover, if Cα,β is non-empty then tα,βu

is a face of the subdivision. The implied association defines the duality. l

We end this section with two examples of tropical hypersurfaces.

Example. Let K � Ctttuu be the field of Puiseux series over C and ω : K Ñ Q8 be

order at 0. Consider the polynomial g P Krx , ys given by g � t x y � t4 x2 � t4 y2 �

pt2� t3�3t6qx�pt2� t5qy�6. The tropicalization Troppgq of g is f � 1x y`4x2`

1.4. TROPICAL GEOMETRY IN HIGHER CODIMENSION 11

4y2`2x `2y `1. The subdivision of the Newton polytope induced by f is drawn in

Figure 1.3(a). Its vanishing locus is depicted in Figure 1.3(b).

4

4

21

1

x2x

xy

y2

y

1

(a) Polyhedral subdivision

(−1, 1)

(1,−1)

(1,−2)

(−2, 1)

xy

x2 x

1

y

y2

x

y

(b) The vanishing locus. The labels on the re-gions denote the monomials of the minimalterms.

Figure 1.3: Tropical vanishing locus of the tropical polynomial 1x y ` 4x2 ` 4y2 `2x ` 2y ` 1 (and the tropical variety of pgq).

Example. Consider SLnpKq, the algebraic variety of n� n-matrices over K of determi-

nant 1. This is a hypersurface of the n2-dimensional affine space MnpKq, the variety of

all n� n matrices with entries in K . Note that the tropicalization of MnpKq is MnpFq,which we represent by the set of F -weighted bipartite graphs on prns, rnsq. A term in

the tropical determinant of a matrix corresponds to a perfect matching in this bipar-

tite graph, and the value of this term equals the weight of the matching. The tropical

vanishing of det X �1 translates to the condition on the bipartite graph that the mini-

mal weight of a perfect matching equals 0, or that it is strictly smaller than 0, in which

case there must be two distinct perfect matchings attaining the minimum weight.

The aims of this section were to establish Theorem 1.3.1 and Theorem 1.3.2. The

existence of a finite tropical basis for a hypersurface is given by Proposition 1.2.2: any

generator of the principal ideal of the hypersurfaces suffices.

1.4 Tropical geometry in higher codimension

We start this section with a number of lemmata about the image of finite subsets of

affine spaces under generic projections. They lead up to a proof of Proposition 1.4.3


that proves that a set is a polyhedral complex if and only if it is so under a suffi-

ciently generic projection. This last proposition is the main tool in reducing the main

theorems of tropical geometry for arbitrary tropical varieties into their hypersurface

specializations. The springboard for proving the main theorems themselves is Theo-

rem 1.4.6.

In this section we refer to affine linear surjective maps Rn Ñ Rm simply as projec-tions. In particular, a projection π is of the form v ÞÑ Av � w, with A an m� n matrix

over R and w P Rm. The kernel kerπ of π is defined to be the kernel of A, i.e. the

set of all v such that πpvq � πp0q � w. Projections are said to be equivalent if they

differ up to an affine automorphism of the range. Since we are mostly interested in

properties of the image of a projection that are invariant under affine automorphisms

(e.g., being a polyhedral complex, dimension) we focus on equivalence classes of pro-

jections. The map π ÞÑ kerπ is a bijection between equivalence classes of projections

and linear subspaces of Rn of dimension n�m.

Let A � Rn be a finite set of affine linear subspaces. It is called complete when

it is closed under intersections. The dimension of A is defined to be the maximal

dimension of its elements. We write |A | for the union of the affine subspaces inA .

A projection π : Rn Ñ Rm is said to be regular with respect toA when it respects

the dimensions of the elements ofA and detects the inclusion of subspaces inA , i.e.

when

(i) dimπpAq � dim A for all A PA ,

(ii) A� B when πpAq � πpBq for all A, B PA .

Note that m ¡ dimA when π is regular with respect to A, unless |A | is an affine

subspace itself. The next lemma shows that regular projections are abundant.

Lemma 1.4.1 LetA � Rn be a finite set of affine subspaces with dimA � d n. Thenthe collection of kernels of projections Rn Ñ Rd�1 that are regular with respect to Acontain a Zariski dense and open subset of Grn�d�1pRnq.

Proof. The proof follows that of Lemma 4.2. of [7]. Let P � |A | be a finite set such

that every affine subspace in A is the affine span of a subset of P. Write VA,p for the

linear subspace of Rn generated by A PA and p P P.

Let π : Rn Ñ Rd�1 is a projection with kerπX VA,p � t0u for all A PA and p P P.

We prove that π is regular. Firstly, π is injective on all of VA,p. Hence dim A� dimπpAqand πppq P πpAq implies p P A. It remains to prove that πpAq � πpBq implies A � B.

Let p1, . . . , pk P P be a subset whose affine span is A. Then πppiq P πpBq for all i Pt1, . . . , ku. Hence pi P B and thus A� B. This proves that π is regular. Thus, the set of

kernel of regular projections contains the set that is the intersection of all sets,

tW P Grn�d�1pRnq |W X VA,p � t0uu, with A PA and p P P.


Since the dimension of VA,p is at most d � 1 each of these sets is Zariski dense in

Grn�d�1pRnq and hence their intersection is. l

Lemma 1.4.2 Let A be a finite set of affine subspaces of Rn with dimA � d n andlet U � Grn�d�1pRnq be a Zariski dense subset. Then there exist d � 1 affine projectionsπ0,π1, . . . ,πm : Rn Ñ Rd�1 with kernels in U such that for every v P |A | there existsi P t0, . . . , mu such that π�1

i pπipvqq X |A | � tvu.

Moreover, for any v0 P |A | one can choose the projections such that for all i Pt0, . . . , du it holds that π�1

i pπipv0qq X |A | � tv0u.

Proof. The proof follows Lemma 4.3. of [7]. In order to make an induction we prove

the following stronger statement. Let A 1 � A , with dimA 1 � d 1. There are d 1 � 1

affine projections π0,π1, . . . ,πd1 in U such that for each v PA 1 there is i P t0, . . . , d 1uwith π�1

i pπipvqq X |A 1| � tvu.

We proceed by induction on d 1. If d 1 � �1, then A 1 � H and the statement is

vacuously true. Suppose that d 1 ¥ 0 and that the statement holds for all integers

smaller than d 1. Since the set of projections regular with respect to A contains a

dense open subset of Grn�d�1pRnq (cf. Lemma 1.4.1) and U is dense there exists a

projection π that is both in U and regular with respect to A . Let X � |A 1| be the

subset consisting of all v P |A 1| for which there exists w P |A | with w � v such that

πpvq � πpwq.

For any affine hyperplane W PA 1 it holds that,

W X X �¤V

pW Xπ�1pπpV qqq, with V PA and W � V ,

by regularity of π. Since π is regular πpW q � πpV q for all W � V and hence the

dimension of W Xπ�1pπpV qq is strictly smaller than that of W . Now set,

A 2 � tW Xπ�1pπpV qq |W PA 1, V PA and W � Vu.

By the induction hypothesis there exist m projections π0, . . . ,πd�1 such that the state-

ment holds forA 2. By construction, adding πd � π yields the statement forA 1. l

Proposition 1.4.3 Let S � Rn be an arbitrary subset. Let U � Grn�d�1pRnq a densesubset such that for every projection π : Rn Ñ Rd�1 with kernel in U the image πpSqis a polyhedral complex of dimension at most d. Then there are projections π0, . . . ,πn :

Rn Ñ Rd�1, with kernel in U, such that,

S �n£

i�0

π�1i pπipSqq. (1.2)


In particular, S is a polyhedral complex.

Proof. The proof consists of two parts. In the first we show that S is contained in the

union of a finite collection of affine subspaces of dimension d. In the second we apply

Lemma 1.4.2 to give the statement.

We construct the collection of affine subspaces by induction. SetA0 � tRnu. Sup-

pose we have constructed a collectionAk of affine subspaces of dimension n� k ¡ dsuch that S � |Ak|. We claim there exists a projection π : Rn Ñ Rd�1 with kerπ P Usuch that kerπ and A affinely span Rn for all A PAk.

Since n� k ¤ d � 1 there exists a finite collection L of d � 1-dimensional linear

subspaces of Rn such that for each A P Ak there is a subspace L P L that is parallel

to an affine subspace of A. Moreover, since the collection of complements of each L is

open and dense in Grn�d�1pRnq and L is finite the collection of subspaces that are

complements to all L PL is open and dense as well. Thus, the intersection of U with

the latter subset is dense, thus in particular, non-empty. Let π be a projection with

kernel in this subset. Then π satisfies the stated criterion. In the second part of the

proof we refer to this specific choice of π by πn�k.

Let B be a finite set of hypersurfaces in Rd�1 such that πpSq � |B |. Set Ak�1 to

be,

Ak�1 � tπ�1pBq X A | B PB and A PAku.

By construction S � |Ak�1|. It remains to show that the elements of Ak�1 have

dimension at most n� k� 1. Suppose that AXπ�1pBq has dimension n� k for some

choice of A and B. Since A has dimension n � k it would follow that A � π�1pBq.However, the affine span of kerπ and A would be contained in π�1pBq, and hence not

equal to Rn. This contradicts the choice of π.

For the second part of the proof, first note that by construction,

|An�d | �n£

i�n�d�1

π�1i pπipSqq.

By Lemma 1.4.2 there exist d � 1 affine projections π0, . . . ,πd such that for all v P|An�d there exists i P t0, . . . , du such that tvu � |An�d | Xπ

�1i pπipvqq|. Thus,

S �An�d Xd£

i�0

π�1i pπipSqq �

n£i�0

π�1i pπipSqq.

This proves (1.2). Since all the intersectands are polyhedral complexes their intersec-

tion is. l

This ends the part on generic projections of arbitrary subsets of Rn. From here on

we apply the results to tropical varieties.


By abuse of terminology, a monomial map Kn Ñ Kd�1 is a map of the form

pa1xα1 , . . . , ad�1xαd�1q, where ai P K and αi P Nn. The tropicalization of a mono-

mial map restricts to an affine linear map Rn Ñ Rd�1 of the form v ÞÑ Av�w, where

A P Nd�n and w P F d�1.

Proposition 1.4.4 Let φ : Kn Ñ Kd�1 be a monomial map and X � Kn an irreduciblealgebraic variety of dimension d. Write Y � Kd�1 for the Zariski closure of φpX q. ThenTrop pφqpTrop pX qq � Trop pY q, and equality holds when Y is a hypersurface.

Proof. We use an elementary argument to show that Trop pφqpTrop pX qq � Trop pY q.Set I � I pX q and J � pφ�q�1pIq, where φ� : Kry1, . . . , yd�1s Ñ Krx1, . . . , xns is the

pullback map induced by φ. Let v P Trop pX q and consider w � Trop pφqpvq. Let f P J .

We show that w is in the tropical vanishing locus of Trop p f q. Write,

f �¸β

cβyβ , Trop pφqpvq � A � v� u.

for A P Nn�pd�1q, u P F d�1 and consider,

Trop p f qpwq �àβ

ωpcβq dwβ

� minβpωpcβq � β � Av� β � uq

� minβppωpcβq � β � uq � pβ � Avqq

� Trop pφ�p f qqpvq

Moreover, the terms occurring in the minimum of the third line and the fourth line are

the same. Since v P Trop pX q, the minimum of those terms in Trop pφ�p f qq is attained

at least twice, or Trop pφ�p f qq equals 8. This shows that w P Trop pY q.

Suppose that Y is a hypersurface and let w P Trop pY q. The set Y Xω�1pwq is

dense in Y , by Theorem 1.3.1. Hence it intersects φpX q. There is p P X such that

ωpφppqq � w. The statement follows by ωpφppqq � Trop pφqpωppqq and the fact that

ωppq P Trop pX q. l

In the next proposition it is ascertained that the projections required to prove the

main theorems are indeed dense, and hence allow us to apply Proposition 1.4.3.

Proposition 1.4.5 Let X � Kn be an irreducible algebraic variety of dimension d. WriteM for the set of monomial maps φ : Kn Ñ Kd�1 that satisfy,

(i) the map Trop pφq : Rn Ñ Rd�1 is surjective,(ii) the Zariski closure of φpX q is a hypersurface.

The set of linear spaces tkerpTrop pφqq | φ PM u is dense in Grn�d�1pRnq.


Proof. We restrict ourselves to the case char K � 0.

Since condition (i) defines a dense subset of Grn�dpRmq it is sufficient to prove

that the maps satisfying condition (ii) define a subset of Grn�dpRnq that contains a

dense open subset.

Denote the restriction of φ to X by φX and write Y for the Zariski closure of the

image of φX . Define U � X by the following expression,

U � tp P X | dppφX q : TpX Ñ TφppqY has rank du.

If φX pUq intersect the nonsingularity locus of Y non-trivially, then Y is a hypersurface

in Kd�1. After all, if p P U and φX ppq is non-singular, then

dim Y � dimTφX ppqY ¥ dim dppφX q�TppX q

�� d.

Let q be a non-singular point in X . We claim that if dqφX has rank d, then there

exists p P U with φX ppq non-singular. Since dqφX has rank d the set U is open and

dense in X . When the characteristic of K is 0 the singularity locus of Y is contained in

a subset of Y of codimension 1. Thus, in that case φpUq intersects the non-singularity

locus of Y . This proves the statement when char K � 0, since for all φ outside of the

zero locus of detdqφX the closure of φpX q is a hypersurface. l

The next theorem is the most important statement in this section. It is a straight-

forward combination of the results on generic projections, the previous proposition

and the fundamental theorem of tropical geometry for hypersurfaces.

Theorem 1.4.6 Let X � Kn be an irreducible algebraic variety of dimension d. Thereexists monomial maps φ0, . . . ,φn : Kn Ñ Kd�1 such that,

Trop pX q �d£

i�0

Trop pφiq�1�

Trop pφipX qq�, (1.3)

where for each i P t0, . . . , nu the Zariski closure φipX q is a hypersurface and Trop pφiq isa projection. In particular, Trop pX q is a polyhedral complex of dimension at most d.

Proof. Let U denote the subset of monomial maps φ : Kn Ñ Kd�1 whose tropi-

calization is surjective and satisfy that the Zariski closure Z of φpX q is a hypersur-

face. Then the set tkerpTrop pφqq | φ P Uu is a dense subset of Grn�d�1pRnq. Then

for every π with kernel in this set there exists φ such that π � Trop pφq. The im-

age πpTrop pX qq equals the tropicalization of Z . By Theorem 1.3.2 it is a polyhedral

complex of dimension d. Hence, by Proposition 1.4.3, there exist monomial maps

φ0, . . . ,φn : Kn Ñ Kd�1 that tropicalize to projections with kernel in U such that

(1.3) holds. l


Corollary 1.4.7 The tropical variety Trop pX q has a tropical basis of cardinality n� 1.

Proof. Let φ0, . . . ,φn be as in the theorem and write I � I pX q. The Zariski clo-

sure Zi of φipX q is a hypersurface, and hence the zero locus of a single polynomial

hi P Kry1, . . . , yd�1s with φ�i phiq P I . We claim that tφ�i phiq P I | i � 0, . . . , nu is

a tropical basis for Trop pX q. By Proposition 1.2.2 the tropicalization Trop pZiq equals

T phiq. Then Trop pφiq�1pTrop pZiqq � T pφ�i phiqq. Applying Theorem 1.4.6 yields that

φ�i phiq form a tropical basis. l

Finally, we prove the fundamental theorem.

Theorem 1.4.8 Let X � Kn be an algebraic variety. Then ωpX q � Trop pX q.

Proof. Let v P Trop pX q. Let φ : Kn Ñ Kd�1 be a monomial map and write π : Rn Ñ

Rd�1 for its tropicalization. Suppose that the closure of φpX q is a hypersurface, that π

is surjective and that Trop pX qXπ�1pπpvqq � tvu. Such φ exist, since the combination

of the first two conditions (cf. Proposition 1.4.5), as well as the last condition, define

subsets of Grn�d�1pRnq containing an open dense set.

Write Z for the Zariski closure of φpX q. By Proposition 1.4.4 it holds that πpX q �Trop pZq. By Theorem 1.3.1 the preimage of πpvq under the valuation ω is dense in Zand hence intersects φpX q non-trivially. Thus, there exists p P X such that πpωppqq �ωpφppqq � πpvq. By the fact that ωppq P Trop pX q and the choice of φ it holds that

ωppq � v, and hence v PωpX q. l

Proposition 1.4.9 Let X � Kn be an irreducible algebraic variety of dimension d. ThenTrop pX q is the underlying set of a pure polyhedral complex of dimension d.

Proof. By Theorem 1.4.6 the set Trop pX q is the underlying set of a polyhedral complex

of maximal dimension at most d. Let P1Y. . .YPk be a non-redundant decomposition of

Trop pX q into polyhedra. Suppose that Trop pX q is not pure. There exists i P t1, . . . , kusuch that dim Pi d. Assume without loss of generality that i � 1.

Let π be (i) a projection Rn Ñ Rd�1 that is the tropicalization of a map φ :

Kn Ñ Kd�1 with the Zariski closure of φpX q a hypersurface, and (ii) regular with

respect to the collection of affine spaces spanned by the Pi . Then πpTrop pX qq is a pure

polyhedral complex of dimension d, since it is the tropicalization of a hypersurface

(cf. Proposition 1.4.4). Hence, πpP1q is contained in the union of πpPiq, with i � 1.

In fact, it is contained in the union of πpPiq with dimpπpP1q X πpPiqq � dimπpPiq

and hence πpP1q and πpP1q X πpPiq span the same affine subspace of Rd�1. Thus,

πpA1q � πpAiq where A j is the affine span of Pj . Hence A1 � Ai by regularity of π.

Since π is injective on Pj and πpP1q is contained in the union of πpPjq, as above, it


holds that,

P1 �k¤

i�2

Pi ,

showing that P1 is redundant in the decomposition. This proves the statement. l

1.5 Tropical geometry and valuation theory

The main theorem of tropical geometry states that the non-Archimedean amoeba

ωpX q of X and the intersection of the tropical zero loci of the polynomials in the ideal

of X coincide, i.e. that the sets ωpX q and Trop pX q coincide. There is another way to

arrive at the same set, by considering the collection of (real-valued) valuations maps

on the coordinate ring of X that extend ω.

Let X � Kn be an embedded algebraic variety. The embedding gives rise to a

surjective homomorphism Krxs Ñ KrX s. We denote the image of f P Krxs under this

homomorphism by f . Write BpX q for the set of all real-valued valuations on KrX sthat extend ω, i.e.

BpX q � tν | ν : KrX s Ñ R8 is a valuation and νpaq �ωpaq for all a P Ku.

By slight abuse of terminology we refer to BpX q as the Berkovich analytic space of X(cf. [51], [31]).

The Bieri-Groves set of the embedding X � Kn is the image of BpX q un-

der the restriction of the valuations to (the monoid generated by) the coordinates

x� t x1, . . . , xnu and denoted BGxpX q, i.e.,

BGxpX q � tpνp x1q, . . . ,νp xnqq | ν PBpX qu � Rn8.

When the embedding is clear from context we write BGpX q. The main theorem of this

section is that BGxpX q X F n and Trop pX q coincide. This was proved in [4], making

use of the techniques of [5]. We prove it directly from the main theorem of tropical

geometry.

Theorem 1.5.1 Let X � Kn be an algebraic variety. Then BGxpX q X F n � Trop pX q.

Proof. Write π : Krxs Ñ KrX s for the canonical projection. We remind the reader that

we write f for πp f q.

Let w P BGxpX q. There exists a valuation ν : KrX s Ñ R8 such that νp x iq � wi .

The pullback π�pνq defined by π�pνqp f q � νp f q is a valuation on Krxs that satisfies

1.5. TROPICAL GEOMETRY AND VALUATION THEORY 19

π�pνqpx iq � wi and π�pνqp f q � 8 for all f P kerπ� I pX q. Write,

f �¸α

cαxα

and suppose the minimum in Trop p f qpwq is attained uniquely by some term α�. Then

π�pνqp f q �ωpcα�qdvα�

, by the properties of valuations, and henceωpcαqdwα �8

for all α. Thus Trop p f qpwq � 8.

The other inclusion is a direct consequence of the fundamental theorem of trop-

ical geometry (Theorem 1.4.8). Suppose w P Trop pX q. There exists p P X such

that ωppq � w. The map g ÞÑ ωpgppqq is a valuation νp : KrX s Ñ R8. It satisfies

νpp x iq �ωpx ippqq �ωppiq � wi . Thus, w P BGxpX q. l

Remark. The Berkovich analytic space is quite a large and complicated set. The main

theorem of tropical geometry essentially states that to obtain the restrictions of valu-

ations to a fixed finite subset of the coordinate ring of the variety, the point valuations

suffice. This is quite a remarkable result.

Chapter 2

Tropical unirational varieties

Tropical geometry has proved useful for implicitization, i.e., for determining equations

for the image of a given polynomial or rational map [61, 62, 64]. The fundamental un-

derlying observation is that tropicalizing the map in a naive manner gives a piecewise

linear map whose image is contained in the tropical variety of the image of the orig-

inal map. Typically, this containment is strict, and for polynomial maps with genericcoefficients the difference between the two sets was determined in [64]. Polynomial

or rational maps arising from applications are typically highly non-generic, and yet

it would be great if those maps could be tropicalized to determine the tropical vari-

ety of their image. Rather than realizing that ambitious goal, this chapter identifies a

concrete research problem and presents several useful tools for attacking it.

The setup in this chapter differs slightly from that of Chapter 1. We use K to

refer to an algebraically closed semifield equipped with a surjective valuation ω onto

a tropical semifield F � R8. However, algebraic varieties are embedded in some

algebraic torus over K and their tropical varieties are underlying sets of polyhedral

complexes in Rn.

More precisely, let Tn denote the algebraic torus Kn� and let X � T n be a subvariety.

Write I for the ideal of X in Krx�1, . . . , x�1s � KrTns and consider the tropicalization

of a polynomial in I to be a polynomial over R8. Then the tropicalization of X is

the intersection of the tropical vanishing loci of the tropicalizations of all elements

of I X Krx1, . . . , xns, intersected with Rn. This definition coincides with the common

definition of the tropical variety of a subvariety of a torus (cf. [52]). Clearly, the main

theorem of tropical geometry does not hold as formulated in Theorem 1.4.8. Instead,

it would read Trop pX q X F n� �ωpX q.

Similarly, the tropicalization of a non-zero polynomial f P Krx1, . . . , xns is a poly-

nomial over R8 with induced map Rn Ñ R. Note that the map induced by the zero

polynomial is not defined in the setting of this chapter. By part (i) of Lemma 1.2.1,

21

22 CHAPTER 2. TROPICAL UNIRATIONAL VARIETIES

(−u,−u)

(−s,−s)u

s

(−u, 0)

(s− u, 0)

(0, u− s)

(0,−s)

Figure 2.1: Outside the lines, Trop pψq is linear and of the indicated form.

we have Tropp f gqpvq � Tropp f qpvq d Troppgqpvq for non-zero polynomials f , g, and

this implies that we can extend the operator Trop to rational functions by setting

Trop p f {gq � Trop p f q � Trop pgq. We further extend this definition to rational maps

φ � p f1, . . . , fnq : Tm 99K Tn by setting Troppφq � pTropp f1q, . . . , Tropp fnqq : Rm Ñ Rn.

Now the central definition of this chapter is the following.

Definition 2.0.1 A subvariety X of Tn is called tropically unirational if there exists a

natural number m and a dominant rational map φ : T m 99K X such that the image

imTrop pφq equals Trop pX q. The map φ is then called tropically surjective.

We recall that the inclusion imTroppφq � Trop pX q always holds (cf. [21]). The

following example shows that this inclusion is typically strict, but that φ can some-

times be modified (at the expense of increasing p) so as to make the inclusion into an

equality.

Example. Let X � T2 be the line defined by y � x �1, with the well-known tripod as

its tropical variety. Then the rational map φ : T1 99K T2, t ÞÑ pt, t � 1q is dominant,

but the image of its tropicalization only contains two of the rays of the tripod, so φ is

not tropically surjective. However, the rational map,

ψ : T2 99K X � T2, ps, uq ÞÑ p1� s

u� s,1� u

u� sq

is tropically surjective. Indeed, see Figure 2: under Trop pψq, the north-west and

south-east quadrants cover the arms of the tripod in the north and east directions,

respectively, and any of the two halves of the north-east quadrant covers the arm

of the tripod in the south-west direction. So X is tropically unirational. There is no

tropically surjective rational map into X with m� 1.

The central question that interests us is the following.

2.1. SOME CLASSES OF TROPICALLY UNIRATIONAL VARIETIES 23

Question 1

Is every unirational variety tropically unirational?

This chapter is organized as follows. In Section 2.1 we review the known fact that

(affine-)linear spaces and rational curves are tropically unirational. In Section 2.2

we prove that, at least for rational varieties, our central question above is equivalent

to the apparently weaker question of whether Trop pX q is the union of finitely manyimages im Trop pφiq, i � 1, . . . , N with each φi a rational map Tpi 99K X . This involves

the concept of reparameterizations: precompositions φ�α of a dominant rational map

φ into X with other rational maps α; since tropicalization does not commute with

composition, Trop pφ �αq may hit points of Trop pX q that are not hit by Trop pφq.

In Section 2.3 we introduce a somewhat ad-hoc technique for finding suitable

(re)parameterizations. Together with tools from preceding sections this technique al-

lows us, for example, to prove that the hypersurface of singular n � n-matrices is

tropically unirational for every n. In Section 2.4 we prove that for X unirational, ev-

ery sufficiently generic point on Trop pX q has a dimpX q-dimensional neighborhood

that is covered by Trop pφq for suitable φ; here we require that K has characteristic

zero. Combining reparameterizations, we find that there exist dominant maps into

X whose tropicalization hit full-dimensional subsets of all full-dimensional polyhedra

in the polyhedral complex Trop pX q. But more sophisticated methods, possibly from

geometric tropicalization, will probably be required to give a definitive answer to our

central question.

2.1 Some classes of tropically unirational varieties

We start with some elementary constructions of tropically unirational subvarieties of

tori.

Lemma 2.1.1 If X is a tropically unirational subvariety of Tn, then so is its imageLuπpX q under any torus homomorphism π : Tn Ñ Tq followed by left multiplicationLu with u P Tq.

Proof. Ifφ : Tm 99K X is tropically surjective, then we claim that so is Lu�π�φ : Tm 99K

Y :� LupπpX qq. Indeed, since φ is a monomial map and Lu is just componentwise

multiplication with non-zero scalars, we have Trop pLu�π�φq � Trop pLuq�Trop pπq�

Trop pφq. Here the first map is a translation over the componentwise valuation ωpuqof u, and the second map is an ordinary linear map. The claim follows from the known

fact that Trop pLuq�Trop pπqpTrop pX qq � Trop pY q, which is a consequence of the main

theorem of tropical geometry (cf. Theorem 1.4.8). l

The following is a consequence of a theorem by Yu and Yuster [71].


Proposition 2.1.2 If X is the intersection with Tn of a linear subspace of Kn, then X istropically unirational.

Proof. Let V be the closure of X in Kn, by assumption a linear subspace. The support of

an element v P V is the set of i such that vi � 0. Choose non-zero vectors v1, . . . , vp P

V such that the support of each vector in V contains the support of some vi . Let

A P Kn�p be the matrix with columns v1, . . . , vp, and let vpAq P pRY t8uqn�p be the

image of A under coordinate-wise valuation. Then by Theorem 16 of [71] states that

Trop pV q � pR Y t8uqn is equal to the image of pR Y t8uqp under tropical matrix

multiplication with ωpAq. This implies that the rational map Tp 99K Tn, v ÞÑ Av is

tropically surjective. l

Another argument for the tropical unirationality of linear spaces will be given in

Section 2.3. There is a more general statement in the context of tropical linear spaces

associated to valuated matroids in Chapter 4.

Lemma 2.1.3 Let X � Tn be a closed subvariety, and write X for the the cone tpt, t pq |t P K�u over X in Tn�1. Then X is tropically unirational if and only if X is.

Proof. If X is tropically unirational, then so is T�X � Tn�1, and hence by Lemma 2.1.1

so is the image X of the latter variety under the torus homomorphism pt, pq ÞÑ pt, t pq.Conversely, if X is tropically unirational, then so is its image X under the torus homo-

morphism pt, pq ÞÑ t�1p. l

Note that Trop pX q � pt0u�Trop pX qq�Rp1, . . . , 1q; we will use this in Section 2.2.

We can now list a few classes of tropically unirational varieties.

Corollary 2.1.4 Intersections with Tn of affine subspaces of Kn are tropically unira-tional.

Proof. If X the intersection with Tn of an affine subspace of Kn, then the cone X is the

intersection with Tn�1 of a linear subspace of Kn�1. Thus the corollary follows from

Proposition 2.1.2 and Lemma 2.1.3. l

The following corollary has been known at least since Speyer’s thesis [58].

Corollary 2.1.5 Rational curves are unirational.

Proof. Let φ � p f1, . . . , fnq : T 99K Tn be a rational map, and let Y be the rational

curve parameterized by it. Let S � K be a finite set containing all roots and poles of

the fi , so that we can write

fipxq � ci

¹sPS

px � sqeis ,

2.1. SOME CLASSES OF TROPICALLY UNIRATIONAL VARIETIES 25

where the ci are non-zero elements of K and the eis are integer exponents. Let X � TS

be the image of the affine-linear linear map T 99K TS given by x ÞÑ px�sqsPS . Then X is

tropically unirational by Corollary 2.1.4. Let π : T S Ñ T n be the torus homomorphism

mapping pzsqsPS to p±

sPS zeiss qi , and let u � pciqi P T n. Then the curve Y is the image

of X under Lu �π, and the corollary follows from Lemma 2.1.1. l

Corollary 2.1.6 The variety in T m�n of m� n-matrices of rank at most 2 is tropicallyunirational.

Proof. Let φ : Tm � Tm � Tn � Tn 99K T m�n be the rational map defined by

φ : pu, x , v, yq ÞÑ diagpuqpx1tr � 1y trqdiagpvq,

where diagpuq, diagpvq are diagonal matrices with the entries of u, v along the di-

agonals; 1t , 1 are the 1 � n and the m � 1 row vectors with all ones; and x , yare interpreted as column vectors. Elementary linear algebra shows that φ is dom-

inant into the variety Y of rank-at-most-2 matrices. Moreover, φ is the composi-

tion of the linear map pu, x , y, vq ÞÑ pu, x1t � 1y t , vq with the torus homomorphism

pu, z, vq ÞÑ pdiagpuqz diagpvqq. Hence Y is tropically unirational by Proposition 2.1.2

and Lemma 2.1.1. l

Corollary 2.1.7 The affine cone over the Grassmannian of two-dimensional vector sub-spaces of an n-dimensional space (or more precisely its part in Tp

n2q with non-zero Plücker

coordinates) is tropically unirational.

Proof. The proof is identical to the proof of Corollary 2.1.6, using the rational map

Tn � Tn ÞÑ Tpn2q, pu, xq ÞÑ puiu jpx i � x jqqi j .

l

Interestingly, Grassmannians of two-spaces and varieties of rank-two matrices are

among the few infinite families of varieties for which tropical bases are known [18].It would be nice to have a direct link between this fact and the fact, used in the

preceding proofs, that they are obtained by smearing around a linear space with a

torus action.

Corollary 2.1.8 The varieties defined by A-discriminants are tropically unirational.

Proof. Like in the previous two cases, these varieties are obtained from a linear variety

by smearing around with a torus action; this is the celebrated Horn Uniformisation[28, 38]. l


In fact, this linear-by-toric description of A-discriminants was used in [20] to give

an efficient way to compute the Newton polytopes of these discriminants in the hy-

persurface case. A relatively expensive step in this computation is the computation of

the tropicalization of the kernel of A; the state of the art for this computation is [55].

2.2 Combining reparameterizations

A fundamental method for constructing tropically surjective maps into a unirational

variety X � Tn is precomposing one dominant map into X with suitable rational maps.

Definition 2.2.1 Given a rational dominant map φ : Tm 99K X � Tn, a reparameter-ization of φ is a rational map of the form φ � α where α : Tp 99K Tm is a dominant

rational map and p is some natural number.

The point is that, in general, Trop pφ�αq is not equal to Trop pφq�Trop pαq. So the

former tropical map has a chance of being surjective onto Trop pX q even if the latter

is not.

Example. In Example 2, the map ψ : ps, tq ÞÑ p 1�su�s

, 1�uu�s

q is obtained from φ : t ÞÑpt, t�1q by precomposing with the rational map α sending ps, uq to 1�s

u�s. Hence ψ is a

tropically surjective reparameterization of the non-tropically surjective rational map

φ.

This leads to the following sharpening of our Question 1.

Question 2

Does every dominant rational map φ into a unirational variety X � Tn have a tropi-

cally surjective reparameterization?

Note that if X is rational and φ : Tm 99K X � Tn is birational, then every dominant

rational mapψ : Tp Ñ X factors into the dominant rational map pφ�1�ψq : Tp 99K Tm

and the map φ. So for such pairs pX ,φq, the preceding question is equivalent to the

question whether X is tropically unirational.

We will now show how to combine reparameterizations at the expense of en-

larging the parameterizing space Tp. For this we need a variant of Lemma 2.1.3.

Let φ � pf1g

, . . . , fn

gq : Tm 99K X � Tn be a dominant rational map where

g, f1, . . . , fn P Krx1, . . . , xms. Let d ¡ 0 be a natural number greater than or equal

to maxtdegpgq, degp f1q, . . . , degp fnqu, and define the homogenizations

g :� xd0 gp

x1

x0, . . . ,

xn

x0q and fi :� xd

0 fipx1

x0, . . . ,

xn

x0q, i � 1, . . . , n.

2.2. COMBINING REPARAMETERIZATIONS 27

These are homogeneous polynomials of positive degree d in n � 1 variables

x0, . . . , xn. The map φ : Tm�1 99K Tn�1 with components p g, f1, . . . , fnq is called

a degree-d homogenization of φ. The components of one degree-d homogenization

of φ differ from those of another by a common factor, which is a rational function

with numerator and denominator homogeneous of the same degree. Any degree-

d homogenization of φ is dominant into the cone X in Tn�1 over X . Recall that

Trop pX q � t0u � Trop pX q � Rp1, . . . , 1q. The following lemma is the analogue of

this statement for imTrop pφq.

Lemma 2.2.1 Let φ be any degree-d homogenization of φ. Then the image of Trop pφq

equals t0u� pimTrop pφqq �Rp1, . . . , 1q.

Proof. For the inclusion �, let v P Rm and let u P R. Setting v :� p0, vq� udp1, . . . , 1q P

Rm�1 and using that g is homogeneous of degree d we find that

Trop p gqpvq � Trop p gqp0, vq � u� Trop pgqpvq � u;

and similarly for the fi . This proves that

Trop pφqpvq � Trop pφqpvq � up1, . . . , 1q,

from which the inclusion � follows. For the inclusion � let v � pv0, . . . , vmq P Rm�1

and set vi :� vi � v0, i � 1, . . . , m. Again by homogeneity we find

Trop p gqpvq � Trop p gqp0, vq � d v0 � Trop pgqpvq � d v0

and similarly for the fi . This implies

Trop pφqpvq � Trop pφqpvq � d v0p1, . . . , 1q,

which concludes the proof of �. l

Lemma 2.2.2 (Combination Lemma) Let φ : Tm 99K X � Tn and αi : Tpi 99K Tm

for i � 1,2 be dominant rational maps. Then there exists a dominant rational mapα : Tp1�p2�1 99K T m such that im Trop pφ � αq contains both imTrop pφ � α1q andimTrop pφ �α2q.

Proof. Consider a degree-d homogenization φ : Tm�1 99K X � Tn�1 and degree-ehomogenizations α1 : Tp1�1 99K Tm�1, α2 : Tp2�1 99K Tm�1 of α1,α2, respectively.

Define α : Tp1�1 � Tp2�1 99K Tm�1 by

αpp, qq � α1ppq � α2pqq.


We claim that

im Trop pφ � αq � imTrop pφ � αiq for i � 1, 2.

Indeed, since φ has polynomial components and since α2 is homogeneous of positive

degree e, we have

φpα1puq �α2pvqq � φpα1puqq � terms divisible by at least one variable v j .

As a consequence, for pµ, νq P Rp1�1 �Rp2�1 we have

Trop pφ � αqpµ, νq �mintTrop pφ � α1qpµq, terms containing at least one ν ju.

Hence if µ is fixed first and ν is then chosen to have sufficiently large (positive)

entries, then we find

Trop pφ � αqpµ, νq � Trop pφ � α1qpνq.

This proves that im Trop pφ � α1q � im Trop pφ � αq. Repeating the argument with the

roles of 1 and 2 reversed proves the claim.

Now we carefully de-homogenize as follows. First, a straightforward computation

shows that φ � αi is a degree-de homogenization of φ � αi for i � 1, 2, hence by

Lemma 2.2.1 we have

impTrop pφ � αiqq � t0u� imTrop pφ �αiq �Rp1, . . . , 1q.

Similarly, writing α � pa0, . . . , amq : T p1�1�p2�1 99K T m for the components of α we

define α : T p1�p2�1 99K T m as the de-homogenization of α given by

αpu, vq ��

a1p1, u, vq

a0p1, u, vq, . . . ,

amp1, u, vq

a0p1, u, vq

.

A straightforward computation shows that φ � α is a degree-de homogenization of

φ �α. Hence by Lemma 2.2.1 we have

impTrop pφ � αqq � t0u� imTrop pφ �αq �Rp1, . . . , 1q.

Now the desired containment

imTrop pφ �αq � im Trop pφ �αiq for i � 1,2

2.3. BIRATIONAL PROJECTIONS 29

follows from

t0u� im Trop pφ �αq � pt0u�Rnq X imTrop pφ � αq

� pt0u�Rnq X imTrop pφ � αiq

� t0u� imTrop pφ �αiq

l

2.3 Birational projections

In this section we show that rational subvarieties of T n that have sufficiently many

birational toric projections are tropically unirational. Here is a first observation.

Lemma 2.3.1 Let X � Tn be an algebraic variety and π : Tn Ñ Td a torus homo-morphism whose restriction to X is birational, with rational inverse φ. Then Trop pπq �

Trop pφq is the identify on Rd .

Proof. Let v P Rd be a point where Trop pφq is (affine-)linear. Such points form the

complement of a codimension-1 subset and are therefore dense in Rd . Hence it suf-

fices to prove that Trop pφqpvq maps to v under Trop pπq. Let y P T d be a point with

ωpyq � v where φ is defined and such that x :� φpyq P X satisfies πpxq � y .

Such points exist because ω�1pηq is Zariski-dense in T d (cf. Theorem 1.4.8). Now

w :� ωpxq equals Trop pφqpvq by linearity of Trop pφq at v and Trop pπqpwq � v by

linearity of Trop pπq. l

For our criterion we need the following terminology.

Definition 2.3.1 Let P � Rn be a d-dimensional polyhedron and let A : Rn Ñ Rd be

a linear map. Then P is called A-horizontal if dim ApPq � d.

Proposition 2.3.2 Let X � Tn be an algebraic variety and π : Tn Ñ Td a torus homo-morphism whose restriction to X is birational, with rational inverse φ. Using the Bieri-Groves theorem, write Trop pX q �

�i Pi where the Pi are finitely many d-dimensional

polyhedra. Then imTrop pφq is the union of all Trop pπq-horizontal polyhedra Pi .

Proof. Let Pi be a Trop pπq-horizontal polyhedron. We want to prove that Trop pφq �

Trop pπq is the identity on Pi . To this end, let v P Pi be such that Trop pφq is affine-

linear at w :� Trop pπqv. The fact that Pi is Trop pπq-horizontal implies that such v are

dense in Pi . To prove that Trop pφqpwq equals v let x P X be a point with ωpvqpxq � vsuch that φ is defined at y :� πpxq and satisfies φpyq � x . The existence of such a

point follows from birationality and the density of fibers in X of the valuation map


(cf. Theorem 1.4.8). Now w equals vpyq by linearity of Trop pπq and v � vpxq �vpφpyqq equals Trop pφqpwq by linearity of Trop pφq at w. Hence Trop pφq � Trop pπq

is the identity on Pi , as claimed. Thus imTrop pφq contains Pi . Since the projections of

Trop pπq-horizontal polyhedra Pi together form all of Rd , we also find that imTrop pφq

does not contain any points outside those polyhedra. l

Corollary 2.3.3 Let X � Tn be a birational variety and write Trop pX q ��

i Pi as inProposition 2.3.2. If for each Pi there exists a torus homomorphism π : Tn Ñ Td that isbirational on X and for which Pi is Trop pπq-horizontal, then X is tropically unirational.

Proof. In that case, there exist finitely many homomorphisms π1, . . . ,πN : Tn Ñ Td ,

birational when restricted to X , such that each Pi is Trop pπ jq-horizontal for at least

one j. Then Proposition 2.3.2 shows that the rational inverse φ j of π j satisfies Pi �

imTrop pφ jq. Now use the Combination Lemma 2.2.2. l

In particular, when all coordinate projections to tori of dimension dim X are bira-

tional the variety X is tropically unirational. This is the case in the following state-

ment.

Corollary 2.3.4 For any natural number n the variety of singular n � n-matrices istropically unirational.

Proof. A matrix entry mi j of a singular matrix can be expressed as a rational

function of all other n2 � 1 entries (with denominator equal to the corresponding

pn� 1q � pn� 1q-subdeterminant). This shows that the map Tn2Ñ Tn2�1 forgetting

mi j is birational. Any pm2 � 1q-dimensional polyhedron in Rm2is horizontal with re-

spect to some coordinate projection Rn2Ñ Rn2�1, and this holds a fortiori for the

cones of tropically singular matrices. Now apply Corollary 2.3.3. l

Corollary 2.3.3 also gives an alternative proof of Corollary 2.1.4 stating that trop-

icalizations of affine-linear spaces are tropically unirational.

Proof. [Second proof of Corollary 2.1.4] Let X be the intersection with Tn of a d-

dimensional linear space in Kn. For each polyhedron Pi of Trop pX q there exists a co-

ordinate projection π : Tn Ñ TI , with I some cardinality-d subset of the coordinates,

such that Pi is Trop pπIq-horizontal. Here we have not yet used that X is affine-linear.

Then the restriction πI : X Ñ TI is dominant, and since X is affine-linear, it is also

birational. Now apply Corollary 2.3.3. l

We continue with an example of a determinantal variety of codimension larger

than one whose unirationality is a consequence of Corollary 2.3.3.

Example. Let V � M4�5pKq be the variety of matrices of rank smaller than or equal

to 3. The ideal of V contains all maximal minors and the dimension of V equals 18.

2.3. BIRATIONAL PROJECTIONS 31

One way to see the latter statement is to write a matrix in V in the following form,

�A BC D

�, A P M3�3, B P M3�2, C P M1�3, D P M1�2.

There are no conditions on A, B and C , while D is uniquely determined by the choice

of A, B and C . The dimension thus equals 3 � 3� 3 � 2� 1 � 3� 18.

Let pmi jq denote the standard coordinate functions on M4�5. We aim to show that

the projection into any subset of X of size 18 is birational. Let z1 � mi, j and z2 � ml,k

be the indices of the coordinate functions left out of the projection. Note that if z1

appears in a maximal minor, in which z2 doesn’t, then z1 is a rational function of the

coordinate functions in the maximal minor. In particular, if z1 and z2 are in different

columns, there exist such maximal minors for z1 and z2 and hence both are rational

in the remaining 18 coordinate function.

The case that z1 and z2 are in the same column requires some calculation. Suppose

without loss of generality that z1 � m3,4 and z2 � m4,5. Then,

0 � m35 det M124,234 �m45 det M123,234 �m15 det M234,234 �m25 det M124,234,

0 � m35 det M124,134 �m45 det M123,134 �m15 det M234,134 �m25 det M134,134

by cofactor expansion of the determinants of the matrices M1234,2345 and M1234,1345.

The set of equations has a unique solution for m35 and m45 when

det M124,134 det M123,134 � det M124,234 det M123,134,

showing that the projection is birational.

We conclude this section with a beautiful example, suggested to us by Filip Cools

and Bernd Sturmfels, and worked out in collaboration with Wouter Castryck and Filip

Cools.

Example. Let Y � T5 be parameterized by ps4, s3 t, . . . , t4q, ps, tq P T 2, the affine cone

over the rational normal quartic. Write X :� Y � Y � T5, the first secant variety.

Writing z0, . . . , z4 for the coordinates on T5, we can describe X as the hyperplane

defined by

det

�� z0 z1 z2

z1 z2 z3

z2 z3 z4

�� z0z2z4 � 2z1z2z3 � z2

1z4 � z0z23 � z3

2

� a� 2b� c� d � e.


x0

x4

x2

a

b

c

d

e

Figure 2.2: The Newton polytope of the Hankel determinant. Only the exponents ofz0, z2, z4 have been drawn.

Trop(πI)-horizontal

Trop(πJ)-horizontal

neither

e ac

bd

Figure 2.3: The tropical variety of the Hankel determinant. Two-dimensional regionscorrespond to the monomials a, b, c, d, e.

This polynomial is homogeneous both with respect to the ordinary grading of

Krz0, . . . , z4s and with respect to the grading where zi gets degree i. Hence its New-

ton polygon is three-dimensional; see Figure 2.2. Modulo its two-dimensional lineal-

ity space the tropical variety Trop pX q is two-dimensional. Intersecting with a sphere

yields Figure 2.3.

Now set I :� t1, 2,3, 4u and J :� t0,1, 2,3u. Then the coordinate projections

πI : T5 Ñ TI , πJ :� T5 Ñ TI are birational, since z0 and z4 appear only linearly in

the Hankel determinant. Let P be a full-dimensional cone in Trop pX q, and let tα,βu

be the corresponding edge in the Newton polygon. Then P is Trop pπIq-horizontal if

and only if α0 � β0 and Trop pπJq-horizontal if and only if α4 � β4. Figure 2.3 shows

that all but one of the cones are, indeed, horizontal with respect to one of these

projections. Let P denote the cone corresponding to the edge between the monomials

b � z1z2z3 and e � z32 . By Proposition 2.3.2 and the Combination Lemma 2.2.2,

there exists a rational parameterization of X whose tropicalization covers all cones

of Trop pX q except, possibly, P. We now set out to find a parameterization whose

tropicalization covers P.

Let v P P. By Lemma 2.2.1 we may assume that v is of the form pv0, v1, 2v1, 3v1, v2q.

We aim to show that there exist two reparameterizations of φ : T4 Ñ X , where

φpu0, u1, v0, v1q � pu0 � v1, u0v0 � u1v1, u0v20 � u1v2

1 , u0v30 � u1v3

1 , u0v40 � u1v4

1 q.

2.4. VERY LOCAL REPARAMETERIZATIONS 33

such that v is in the image of at least one of them. Note that from the defining in-

equalities of P it follows that v0 ¥ 0 and v2 ¥ 4v1. Let i P K be a fourth root of unity

and consider the map ψ : T3 Ñ T 4 defined by

ψpx0, x1, x2q ��1� x0,�1, i x1,�x1p1� x2 x�4

1 q�.

A short computation shows that the restriction of the tropicalization of φ � ψ to

the cone defined by w0 ¥ 0, w2 ¥ 4w1 and w2 ¤ w0 � 4w1 is the linear function

pw0, w1, w2q ÞÑ pw0, w1, 2w2, 3w1, w2q. If v satisfies v2 ¤ v0 � 4v1 then the image of

pv0, v1, v2q under this tropicalization is exactly v.

If v satisfies v2 ¥ v0�4v1 it is in the image of the tropicalization of φ�ψ�ι, where

ιpx0, x1, x2q � px�10 , x�1

1 , x�12 q.

The tropicalization is linear on the cone 0 ¥ w0, 4w1 ¥ w2 and w2 ¥ w0 � w1 and

maps �pv0, v1, v2q to v.

2.4 Very local reparameterizations

Let X � Tn be a d-dimensional rational variety that is the closure of the image of a

rational map φ : Tm 99K Tn. Suppose without loss of generality that X is defined over

a valued field pK ,ωq such that ωpK�q is a finite dimensional vector space over Q.

Write v for the image of v P R under the canonical projection RÑ R{ωpK�q. We can

now state the main result of this section and its corollary.

Theorem 2.4.1 Assume that the field K has characteristic zero. Let v P Trop pX q and setd to be the dimension of the Q-vectorspace spanned by v1, . . . , vn. There exists a rationalmap α : Td 99K Tm and an open subset U � Rd such that the restriction of Trop pφ �αq

to U is an injective affine linear map, whose image contains v.

Corollary 2.4.2 Assume that K has characteristic zero. Let tP1, . . . , Pku be a finite set ofωpK�q-rational polyhedra of dimension dim X such that

Trop pX q �k¤

i�1

Pi .

There exist a natural number p and a rational map α : Tp 99K Tm such that the image ofTrop pφ �αq intersects each Pi in a dim X -dimensional subset.

Proof. By the theorem, for each polyhedron Pi there exists a reparameterization αi

such that the tropicalization of φ �αi hits Pi in a full dimensional subset. They can be


combined using the Combination Lemma. l

The main step in the proof of the theorem is Proposition 2.4.5, which is a valuation

theoretic result.

Let v be a point of Rn. Such a point defines a valuation ωv on the field of rational

functions L � Kpy1, . . . , ynq of T n by

ωvphq � Trop phqpvq, h P L

Let Lv denote the completion of L with respect to ωv and denote its algebraic closure

by Lv . That closure is equipped with the unique valuation whose restriction to Lv

equals ωv [66, §144]. Denote by KryQ1 , . . . , yQ

n s the subring of Lv generated by all

roots of the elements y1, . . . , yn.

The next lemmata deal with the case n � 1. They allow us to prove Proposi-

tion 2.4.5 below by means of induction on the number of variables.

Lemma 2.4.3 Let v P R such that v is not in the Q-vector space spanned by vpK�q.Then Krt�1, ts is dense in Kptqv .

Proof. Let p{q P Kptq. If q is a monomial we are done, since then 1{q P Krt, t�1s.

Suppose it isn’t. Write,

q �¸

i

qi ti , with qi P K�.

Then the valuation of q equals ωvpqq � miniωpqiq � iv. Moreover, the minimum is

attained exactly once, since otherwise v would be a Q-multiple of some element of

vpK�q. Say it is attained at j. Compute,

p

q�

p

a j t j � pq� a j t jq

� pa j tjq�1 p

1� pq� a j t jq{pa j t jq

� a�1j t� j p

8

n�0

� pq� a j tjq

a j t j

n

The convergence of the power series with respect to ωv is a consequence of ωvpq�a j t

jq ¡ ωvpa j tjq. The limit is easily seen to coincide with p{q. This completes the

proof. l

Lemma 2.4.4 Suppose K is algebraically closed of characteristic 0. Then KrtQs is densein Kptqv .

2.4. VERY LOCAL REPARAMETERIZATIONS 35

Proof. Denote the residue field of Kptqv by k. Note that it is also the residue field of

Kptq under v, by the conditions on v.

We prove by induction on d that all zeroes in Kptqv of a polynomial of degree dover Kptqv can be approximated arbitrarily well with elements of KrtQs. For d � 1

this is the content of Lemma 2.4.3. Assume that the statement is true for all degrees

lower than d. We follow the proof of [67, §14, Satz].Let PpSq � Sd � ad�1Sd�1� . . .� a0 P KptqvrSs. After a coordinate change replac-

ing S by S � 1d

ad�1 we may assume that ad�1 � 0. Indeed, a root s of the original

polynomial can be approximated well by elements of KrtQs if and only if the root

s� 1d

ad�1 can be approximated well, since ad�1 itself can be approximated well.

If now all ai are zero, then we are done. Otherwise, let the minimum among the

numbers vpad�iq{i be ω� qτ, where ω P vpK�q and q PQ, and let c be a constant in

K with valuation ω. Setting S � c tqU transforms P into

cd tdqpUd � bd�2Ud�2 � . . .� b0q,

where each bi is an element of Kpt1{pqv of valuation at least zero, with p the denom-

inator of q. Moreover, some bi has valuation zero. Let QpUq denote the polynomial in

the brackets. The image of QpUq in the polynomial ring krUs over the residue field is

neither Ud as bi has non-zero image in L, nor a d-th power of an other linear form

as the coefficient of Ud�1 is zero. Hence the image of QpUq in krUs has at least two

distinct roots in the algebraically closed residue field k, and therefore factors over kinto two relatively prime polynomials. By Hensel’s lemma [67, §144], Q itself factors

over Kpt1{pqv into two polynomials of positive degree. By induction the roots of these

polynomials can be approximated arbitrarily well by elements of KrtQs, hence so can

the roots of Q and of P. l

Proposition 2.4.5 Let pK ,ωq be an algebraically closed field of characteristic 0 withvaluation ω and v P Rn whose entries are Q-linearly independent over R{ωpK�q. ThenKryQ

1 , . . . , yQn s is dense in Lv .

Proof. Follows from Lemma 2.4.4 by induction on the number of variables. l

We are now ready to prove the main result.

Proof. [Proof of Theorem 2.4.1] Choose τ1, . . . ,τd P R such that their projections in

R{ωpK�q form a basis of the Q-vectorspace spanned by ξ1, . . . , ξn. Let t1, . . . , tn be

variables and denote by L the field Kpt1, . . . , tnq equipped with the unique valuation

ν that extends ω and satisfies νpt iq � τi .

There exists a point x 1 P T mpLξq such that wpφpx 1qq � ξ. By Proposition 2.4.5

there exists an approximation x P T mpKrtQ1 , . . . , tQd s) of x 1 that satisfies wpφpxqq � ξ.

Choose e P N such that every coefficient of x is already in Krt�1{e1 , . . . , t�1{e

d s and set


si � t1{ei . Thus, x P T mpKrs�1 , . . . , s�d sq, and hence defines a rational map T d Ñ T m.

Denote this map α. We show that there exists a neighborhood of σ � 1epτ1, . . . ,τdq

such that the restriction of Trop pφ �αq to U satisfies the conclusions of the theorem.

First, note that, by construction of α, Trop pφ � αqpσq � ξ. Every component

φipxq � φipαps1, . . . , sdqq of φ is a Laurent polynomial over K in the s j with a unique

term zisbi,1

1 � � � sbi,d

d of minimal valuation ξi � vpziq �°d

j�1 bi, jσ j . If we let σ vary in a

small neighborhood and change the valuations of the si accordingly, then for each ithe same term of φi has the minimal valuation. Hence Tpφ �αq is linear at σ with dif-

ferential the matrix pbi jq. Finally, as the numbers η1, . . . ,ηn span the same Q-space as

σ1, . . . ,σd modulo vpK�q, the matrix pbi jq has full rank d. This completes the proof.

l

2.5 Concluding remarks

The concept of tropical surjectivity of a rational map seems natural and concrete,

and, as far as we know, not to have been studied before. This chapter presented

some methods of determining whether a rational map is tropically surjective, and

aims to be a starting point for further study. In particular, the question whether every

unirational variety is tropically unirational is still open. It seems likely that techniques

from geometric tropicalization will prove useful in making further progress on this

question.

Chapter 3

Algebra of idempotent

semirings

In this chapter the focus is moved away from tropicalization map to the tropical semi-

fields. It gathers known results on the broader class of semirings whose addition is

idempotent, into which the tropical semifields fit naturally, and adapts them to our

specific notation and setting.

The chapter serves two purposes. First, a number of results on tropical linear

spaces, the subject of investigation in Chapter 4, hold for arbitrary modules over

idempotent semirings, and they are better stated and proved in this context. Secondly,

it serves to understand the algebraic properties that are necessary for the notion of a

matroid, and hence a tropical linear space, over a semiring to be well-behaved.

3.1 Modules over semirings and homomorphisms

The most general type of algebraic structure considered in this thesis is the semiring,

and its associated structure, the module over a semiring. We start off by defining the

former.

Definition 3.1.1 A semiring is an algebraic structure pR,�, �, 0Rq consisting of a set

R and two binary operations, addition, denoted �, and multiplication, denoted �,

satisfying the conditions

(i) pR,�, 0q is a commutative monoid with identity element 0 P R,

(ii) pR, �, 1q is a monoid with identity element 1 P R,

(iii) multiplication distributes over addition,

(iv) for all x P R it holds that 0 � x � 0 � x � 0,

(v) 0� 1.

37

38 CHAPTER 3. ALGEBRA OF IDEMPOTENT SEMIRINGS

A commutative semiring is a semiring in which multiplication is commutative. If each

element in Rzt0u has a multiplicative inverse the commutative semiring is said to be

a division semiring. A commutative division semiring is called a semifield. In case of

possible ambiguity we write 0R, respectively 1R, for the identity elements of R.

Note that the absorption axiom stating that 0 � x � x �0� 0 for all x P R is implied

by the other semiring axioms when additive inverses always exist. The latter property

is necessary in the implication, which is why the absorption axiom is explicitly added

here.

It is clear from the definition that rings are special cases of semirings. In fact they

are the semirings whose every element has an additive inverse. Idempotent semir-

ings are in some sense on the other end of the spectrum. Their addition operation

is idempotent and hence non-invertible. In formal terms, a semiring pR,�, �,8q is

said to be idempotent when for all x P R it holds that x � x � x , or equivalently,

when 1R � 1R � 1R. The terminology extends to commutative semirings and semi-

fields. Idempotent semirings are also known as dioids, where dioid stands for double

monoid.

Remark. We shall make no distinction between the algebraic structure and the under-

lying set, when reference to the operations is clear from the context. Thus, R denotes

the underlying set, as well as the semiring.

In this thesis all semirings are understood to be commutative, and hence we usually

omit this adjective.

Example. An important example of a semiring that is not a ring is the set of extended

real numbers RY t8u in which x ` y � mintx , yu and x d y � x � y . It is in fact

a semifield, and we refer to it as the real tropical semifield. The symbols ` and d

serve to clearly distinguish the semiring operations from the usual (partially defined)

addition and multiplication on RYt8u. We denote the real tropical semifield by R8.

The terminology and notation extends to subgroups of pR,�q. For example, the

subsemiring of R8 consisting of Q and 8 is referred to as the tropical rational semi-field, or simply the tropical rational numbers, and denoted Q8.

Definition 3.1.2 Let R be a (commutative) semiring. A module M over R is an alge-

braic structure consisting of a set M and the operations of addition M � M Ñ M ,

denoted �, and scalar multiplication R� M Ñ M , denoted �, such that the following

conditions are satisfied,

(i) pM ,�, 0M q is a commutative monoid,

(ii) x � pv�wq � x � v� x �w for each x P R and v, w P M ,

(iii) px � yq � v � x � v� y � v for each x , y P R and v P M ,

(iv) 1R � v � v for all v P M .

3.1. MODULES OVER SEMIRINGS AND HOMOMORPHISMS 39

As in the case of semirings, M is said to be an idempotent module if it has idempotent

addition, i.e. m� m � m for all m P M . Note that the axioms ensure that a module

has a zero element, denoted 0M , or simply 0, when reference to the module under

consideration is clear from the context.

Remark. Modules over semirings are often referred to as semimodules in the litera-

ture. In this thesis, we prefer the term module, for two reasons: (i) the axiom system

of a semimodule is precisely that of a module once one replaces the term ring by

semiring and hence the concept of a semimodule is an extension of that of a module

(this argument also appears in a footnote in [36]), (ii) there is no specific reference

in this thesis to modules over rings, and hence, once one keeps that in mind, there

should be no loss in clarity as a consequence of our naming convention. Moreover, it

saves writing the morpheme semi many times.

Examples. The following are examples of modules over semirings. Many more exam-

ples are to be found in [29].(i) For R a semiring and X a set denote by RX the collection of maps X Ñ R. This

is an R-module under the operations of pointwise addition and scalar multi-

plication. Moreover, such R-modules are free in the category of R-modules and

R-module homomorphisms.

(ii) Let R be the tropical rational semifield and consider the module M over R whose

underlying set is that of tropical real numbers, RY t8u, with module addition

v ` w � mintv, wu and scalar multiplication x d v � x � v. That the module

postulates are satisfied is a consequence of the tropical semifield structure on

RY t8u. It is clear that M is not finitely generated over R, since the quotient

R{Q of additive abelian groups is not finite.

(iii) Every semiring is a module over itself. Addition is the ordinary addition opera-

tion in the semiring and scalar multiplication is multiplication in the semiring.

In this sense, modules over semirings are more general than semirings.

A module over a semiring that has idempotent addition is clearly idempotent itself,

since v � v � p1R � 1Rq � v � v, but in general the implication is not an equivalence.

A simple example is a module M with idempotent addition, whose ring of scalars

acts trivially, i.e. each scalar acts as the endomorphism v ÞÑ 0M . However, if each

multiplication with a scalar defines a different endomorphism (such modules are said

to be faithful), then the implication is an equivalence.

A useful device for the analysis of modules whose addition is idempotent is the

notion of the natural preorder on the module. Since this preorder depends only on the

additive structure of the module it is most naturally defined for general semigroups.

Let S be such a semigroup. The natural preorder relation P� on S is defined by x P� ywhen there exists z P S such that x � y � z. That this actually defines a preorder is


part (i) of the following proposition. For us, part (iii) of the same proposition is the

most important part. It states that when addition on the module is idempotent it is

uniquely determined by P�. Moreover, in that case P� is a meet semi-lattice order,

and we denote it by ¤�, or ¤, when the additive structure under consideration is

clear from the context. Then it is also referred to as the natural order [29], inducedorder or canonical order [43] on the semigroup.

Proposition 3.1.1 Let S be a commutative semigroup. The following statements hold.(i) The relation P� is a preorder,

(ii) If S is a group then P� � S� S,(iii) If S is idempotent then px , yq P P� is equivalent to x � y � x and P� defines a

meet semi-lattice. Moreover, x � y � x ^ y, where ^ denotes the meet operationof P� and the unique maximum of P� is the identity element of S, if it exists. Anymeet semi-lattice is obtainable in this way.

Proof. (i) We need to show that P� is reflexive and transitive. Let x P S. Then x�0� xand hence px , xq P P�. Let x , y, z P S such that px , yq P P� and py, zq P P�. There exist

u, v P S such that x�u� y and y�v � z. Thus, x�pu�vq � z and hence px , zq P P�.

(ii) Let x , y P S. Then x � py � xq � y and hence px , yq P P�.

(iii) Suppose that x � y � x . Then clearly xP� y . Now suppose that x P� y . There

exists z P S such that x � y�z. Then x� y � y�z� y � y�z � x , which shows the

required equivalence. Since P� was already proved to be a preorder in (i) it remains

to show that it is antisymmetric. Let x , y P S and suppose that both xP� y and yP�x .

Then x � x � y � y .

The partial order P� is a lower semilattice order exactly when each set tx , yu has

a unique infimum inftx , yu. We prove that inftx , yu exists and equals x � y , in two

steps. For clarity we denote P� by ¤.

(a) The identity x � px � yq � y � px � yq � x � y implies px � yq ¤ x and

px � yq ¤ y , as required. Hence x � y is a lower bound of tx , yu(b) Suppose that z is a lower bound of tx , yu. Then z � px � yq � pz � xq � y �

z� y � z. Hence, z ¤ px� yq, showing that x� y is the largest lower bound of tx , yu.l

Remark. The correspondence of Proposition 3.1.1 restricts to a correspondence be-

tween idempotent commutative monoids and lower semilattices with a unique maxi-

mal element. The equations defining the unit element of the monoid translate directly

into the required inequalities for the maximal element. Moreover, the correspondence

is (trivially) functorial in the sense that maps that preserve the structure of the in-

duced meet semi-lattice are semigroup homomorphisms, and vice versa.

Note that commutative idempotent semigroups are also known as commutative

bands, see [16].


The remainder of this section is concerned with modules that have idempotent

addition. Properties of such modules are often easier expressed in terms of the canon-

ical order. The next theorem serves to prove that the order restricted to some subset

of the semiring is a lattice order. For continuous semi-lattices this is automatic and

the subset is the entire ring, since the supremum of a set is just the infimum of its

upper bounds. Since we do not assume continuity, more work needs to be done. A

by-product of the proof is that the multiplicative inverse is an anti-automorphism of

lattices on R�, where R� denotes the group of elements of R that have a multiplicative

inverse.

Proposition 3.1.2 Let R be an idempotent semiring. If R� is closed under � then thenatural order on R restricts to a lattice order on R�.

Proof. Let x , y P R�. It is sufficient to prove that x�1 ^ y�1 � px _ yq�1. Denote

x�1 ^ y�1 by w. We show that w�1 � x _ y , in two steps.

(i) To prove that x ¤ w�1 compute,

w�1 � x � w�1 � p1� pw � xqq

� w�1 � p1� py�1 � xqq

� w�1 � px�1 � y�1q � x

� x

Hence, x ¤ w�1, and similarly, y ¤ w�1.

(ii) For the second step, let z P R satisfy x ¤ z and y ¤ z. Then,

w�1 � z � w�1 � p1� pw � zqq

� w�1 � p1� ppx�1 � zq � py�1 � zqqq

� w�1 � x�1 � px � z� px � y�1 � zqq

� w�1 � p1� py�1 � zqq

� w�1 � y�1 � py � zq

� w�1.

Hence, w�1 ¤ z, thereby proving the statement.

l

Corollary 3.1.3 Let F be an idempotent semifield. The natural order on F is a latticeorder.

Proof. It follows from the lemma that the natural order restricted to F� � Fzt0u is a

lattice order. Since x �8 � x for all x P F the element 0 is the top element of the


natural order on F . Thus, x ^ 0� x and x _ 0� 0 for all x P F . l

We continue the discussion of idempotent semirings by distinguishing the class of

tropical semirings in terms of algebraic conditions on semirings. A division semiring

F is Archimedean if for all x , y P F� the statement xZ ¤ y implies x � 1, i.e. when

the only element all of whose powers are smaller than any given invertible element

is the multiplicative identity. Recall that the tropical real numbers are the semifield

with underlying set RYt8u and operations x ` y �mintx , yu, x d y � x � y . Note

that their natural order is the common order on the extended real numbers. The next

theorem is essentially a theorem by Hölder. The statement and its proof, formulated

in the language of lattice ordered groups is found for example in [60].

Theorem 3.1.4 Let F be an idempotent semifield for which the induced order is linearthat satisfies the Archimedean property. Then F embeds into pRYt8u, min,�q.

Proof. By Theorem 2.3.10. of [60] a any lattice ordered group embeds into the lattice

ordered group pR,�q if and only if it is Archimedean and linearly ordered. It is easy

to see that pF�, �q under the natural order satisfies these properties and hence that it

embeds in pR,�q. The embedding extends to an embedding F Ñ R8. l

A tropical semiring is defined to be a subsemiring of an Archimedean idempotent

semifield. The above theorem shows that tropical semirings in our sense coincide with

the tropical semirings as defined in Chapter 1 and, for example, in [53]. In certain

types of statements the Archimedean property is not necessary and in these cases it is

more natural to focus on linearly ordered semirings.

Example. The smallest example of a tropical semiring is the two element semiring

ta, bu with addition and multiplication defined by the tables,

� a b

a a ab a b

� a b

a a bb b b

This semiring is commonly referred to as the Boolean semiring or trivial semifield and

denoted B. The embedding B ãÑ R8 that establishes it as a tropical semiring is a ÞÑ 0

and b ÞÑ 8.

A simple example of a linearly ordered semiring that is not tropical, consider the

three element semiring, R � t�8, 0,8u. It is an extension of the Boolean tropical

semiring t0,8u � R8, with addition defined �8� x � �8 and �8 � x � �8 for

all x P R.

The general notion of an idempotent module has the disadvantage that infima

of infinite sets do not exist in general. An approach often taken in the literature (e.g.


[13]) is to extend the module under consideration to one that is complete in the sense

that every set has an infimum. For our purposes this condition is too strong. After all,

it is not satisfied by the tropical real numbers, since the infimum of the subset R of

R8 does not exist. Requiring that infima of bounded subsets exist will not do, since

we also wish to consider modules over the tropical rational numbers. Instead we will

require infima of very specific subsets of semirings to exist.

Let M be a module over an idempotent semiring R. Such a module is said to be

dually residuated if for each v, w P M with w � 0M the set tx P R | v ¤ x � wu has a

unique minimum. It is then denoted by v : w, or pv : wq, and called the dual residuumof v and w. In terms of the semimodule structure pv : wq is the (infinite) sum of all

elements x P R such that v � x � w � v. The notion of (dual) residual in modules

over semirings is found in [13]. It is well-studied in case the module is the ground

semiring with scalar multiplication given by semiring multiplication. See for example

[70, 65] and the more recent survey [37].

The next statement serves to single out relations between modules that preserve

the property of dual residuatedness .

Proposition 3.1.5 Let R be an idempotent semiring. The following statements hold.

(i) Let M be a module over R and N � M a submodule. If M is dually residuated overR then N is dually residuated over R.

(ii) Suppose that the induced order of R is a lattice order. If R is dually residuated thenany free module of finite rank over R is dually residuated over R.

(iii) Let A be an algebra over R and M a module over A. If A is dually residuated over Rand M is dually residuated over A then M is dually residuated over R.

Proof. We prove the claims in order stated.

(i) This is clear since both the natural order and the action of R on N are the

restrictions of those on M .

(ii) Let M be a free module over R of finite rank. Fix a finite set of free generators

G � M . Let v, w P M with w � 0M and define px gq and pygq in R such that,

v �¸gPG

x g � g, w �¸gPG

yg � g.

Set z � suptpx g : ygq | g P Gu. We claim that pv : wq � z. By a straightforward

computation,

z �w �¸gPG

pz � ygq � g ¥¸gPG

x g � g � v,

which shows that z ¥ pv : wq.


Let z1 P R satisfy v ¤ z1 � w. Then by the premise that M is free and the following

computation,

v � v� z1 �w �¸gPG

px g � z1 � ygq � g,

it holds that x g � z1 � yg � x g for all g P G. In terms of the induced order on R this

states that z1 � yg ¥ x g for each g P G. Hence, z1 ¥ z and the claim follows.

(iii) Let v, w P M with w � 0M . Subscripts A and R denote the dual residua in

the A-module M and the R-module A, respectively, e.g. pv : wqA stands for the dual

residuum of v by w in M over A. Set x � ppv : wqA : 1AqR. We claim that x � minty PR | v ¤ y �wu.

Since v ¤ pv : wqA �w ¤ px �1Aq �w � x �w, it holds that x is at least the minimum.

Suppose that y P R satisfies v ¤ y � w. We need to show that x ¤ y . Multiplying both

sides of the inequality with 1A yields that v ¤ py � 1Aq � w and thus y � 1A ¥ pv : wqA.

Hence y ¥ ppv : wqA : 1Aq � x . This proves the claim. l

Remark. In part (ii) of the preceding proposition the module is required to be of finite

rank since otherwise there is no guarantee on the existence of lowest upper bounds

of infinite subsets of R. Vice versa, if the induced order on R has the property that it

is a lattice that is complete as an upper semi-lattice then the finite rank condition is

superfluous. Since tropical linear spaces, the main geometric objects in Chapter 4, are

finitely generated, it is more natural to require finiteness, instead of completeness.

Remark. Naively one might expect that the dual residuation property of semirings

transfers to finitely generated modules over such semirings. This is not true, since

there might not be a system of generators whose mutual dual residua with respect

to R exist. A simple example is the Q8-submodule of R8 generated by t0, vu, with

v P R8zQ8. The dual residuum pv : 0q does not exist since the set tx P Q8 | v ¤ xuhas no infimum.

When the module has a finite generating set whose mutual dual residua exist it is

dually residuated over its base idempotent semiring, when the induced order of the

base ring is a lattice order. The proof is nearly as the same as for free modules of finite

rank, as in part (ii) of the Proposition 3.1.5.

Dual residuation is a device that allows one to translate order theoretic properties

of idempotent semirings to finitely generated modules over them. The key is the fact

that any element of such a module has a canonical representation as a linear combi-

nation of generators, where each coefficient is a dual residuum . More precisely, let

M be such a module over an idempotent semiring R and v P M . Fix a finite set of


generators G � Mzt0Mu. Then,

v �¸gPG

pv : gq � g. (3.1)

We refer to this sum as the canonical representation of v in terms of G. Similarly, in

this setting pv : gq is referred to as the canonical coordinate of v with respect to g.

Proposition 3.1.6 Let R be an idempotent semiring having induced lattice order and Ma finitely generated module over R that is residuated over R. Then M has induced latticeorder.

Proof. Let v P M . Then v � v � p1� 1q � v � v. By part (iii) of Proposition 3.1.1 the

induced order on M is a meet semi-lattice. It remains to prove that it is also a join

semi-lattice. Here we require that M is finitely generated.

Fix a finite set of generators G � Mzt0Mu of M . Let v, w P M and define u P M by,

u�¸gPG

�pv : gq _R pw : gq

�� g.

We claim that u� v_M w. For each g P G the join pv : gq_R pw : gq is an upper bound

of both pv : gq and pw : gq. Hence u¥ v_M w.

Let u1 be an upper bound of both v and w. Since residuation is increasing in the

first argument both pu1 : gq ¥ pv : gq and pu1 : gq ¥ pw : gq. Thus pu1 : gq is at least the

smallest upper bound pv : gq_Rpw : gq and hence u1 ¥ u. This proves that u¤ v_M w.

l

Example. This example serves to direct the reader’s attention to the fact that finite

generatedness of modules over semirings is a subtle property. Even submodules of free

modules of finite rank over a linearly ordered semifield need not be finitely generated,

as the example in the proof of Proposition 2.2 in [15] shows. We repeat the relevant

part here.

Consider the free module R38 over the real tropical semiring R8 and set vn �

p� 1n, 0, 1

nq for all n P N. Denote by Vn the submodule generated by v1, . . . , vn. Clearly,

Vn � Vn�1 for all n P N. However, Vn�1zVn �H and the minimal module that contains

all vn is not finitely generated.

The last part of this section deals with homomorphisms of idempotent semimod-

ules. There is a notion of (dual) residuation of an increasing map which is an exten-

sion of that of a module (cf. [8]). Dual residuation in a module translates then to dual

residuation of the map R Ñ M defined by x ÞÑ x � w, where w P M . An increasing

map φ : M Ñ N between modules over idempotent semirings is said to be dually


residuated if there exists a map such ψ : N Ñ M such that

φ �ψ¥ idN and ψ �φ ¤ idM , (3.2)

in which the partial order on the collection of maps is the pointwise order. We then

denote the map ψ by φ^. This notation comes from the fact that ψ is the map that

assigns to each w P N the infimum of the preimage of the principal filter with minimal

element w under φ, if it exists. Since this is conceptually important we state the

equivalence in the next lemma. The proof of this lemma is to be found for example in

[8, p.10-11]. We remind the reader that a principal filter is a set of the form ty P M |

x ¤ yu, for some x P M .

Lemma 3.1.7 Suppose φ : M Ñ N is a homomorphism of modules over an idempotentsemiring. Let ψ be a map N Ñ M satisfying (3.2). Then ψpwq � mintv P M | φpvq ¥wu. In particular, the preimage of every principal filter under φ is again a principal filter.

Remark. An increasing map that has a dual residual map is referred to as dually

residuated . This is standard terminology (cf. [8]). It is inconvenient that the prime

property we are interested in is not residuatedness, but its order-theoretic dual. The

reason is essentially the fact that we have defined the natural order as we have, i.e.

v ¤ w if and only v � w � v. This convention is known as the min-convention in

tropical geometry, and its opposite is the max-convention. Had we chosen the max-

convention, the prime property would be residuatedness. Note that the min- and max-

conventions make no difference for the algebra of idempotent semirings, since order-

theoretic statements on the canonical order translate into statements on the additive

structure of the semiring.

The next statement shows that in the category of finitely generated modules that

are residuated over a fixed base semifield, residual maps always exist. The proof is

adapted to our setting from Theorem 5.2. of [8]. The statement below is fairly general

and somewhat technical.

Theorem 3.1.8 Let φ : M Ñ N be an arbitrary map between modules over an idempo-tent semiring R that respects the action of R. If (i) M is finitely generated, (ii) N is duallyresiduated and (iii) φ is a homomorphism of modules satisfying φpMzt0Muq � Nzt0Nu,then φ is dually residuated . In that case,

φ^pwq �¸gPG

pw : φpgqq � g, w P N , (3.3)

where G � Mzt0Mu is a finite set of generators of M.


Vice versa, if φ is dually residuated then it is a homomorphism of modules. If, more-over, when the induced orders on both M and N are lattice orders, then φ^ is a homo-morphism of the induced join semi-lattice, i.e., φ^pv _ wq � φ^pvq _ φ^pwq for allv, w P N, that respects the action of R.

Proof. Let w P N . We need to show that the infimum of tv P M | φpvq ¥ wu exists.

We claim that it equals°

gPGpw : φpgqq � g. Denote this quantity by u. Then φpuq �°gPGpw : φpgqq � φpgq ¥ w by definition of pw : φpgqq. Thus, u is larger than the

infimum. For the other inequality, let u1 P M such that φpu1q ¥ w. Write u1 �°

gPG x g �

g, for certain x g P R. Then x g �φpgq ¥ w and hence pw : φpgqq ¤ x g . Thus, u1 ¥ u.

This shows that u is a lower bound of tv P M | φpvq ¥ wu and hence that φ is dually

residuated with residual map w ÞÑ°

gPGpw : φpgqq � g.

Suppose that φ : M Ñ N is a dually residuated map. Let v, w P M . We need to

show that φpv � wq � φpvq � φpwq, or in order theoretic terms, that φpv ^ wq �φpvq ^φpwq. Since v ^ w is a lower bound of tv, wu and φ is an increasing map it

follows that φpv^wq ¤ φpvq ^φpwq.

Let u be a lower bound of tφpvq,φpwqu. Then φ^puq ¤ φ^pφpvqq ¤ v by (3.2)

and similarly, φ^puq ¤ w. Thus, φ^puq ¤ v ^ w. Thus, u ¤ φpφ^puqq ¤ φpv ^ wq,again by (3.2). Thus, φpv^wq is the largest lower bound of tφpvq,φpwqu and hence

φpv�wq � φpvq �φpwq.

It remains to prove that φ^ respects the join semi-lattice structures on N and

M , if they exist. Let w1, w2 P N . Set R � tv P M | φpvq ¥ w1 _ w2u and Si �

tv P M | φpvq ¥ wiu, for i � 1,2. Let v P R. Then φpvq ¥ w1 _ w2 ¥ wi and

hence v P Si . Thus R � Si and inf R ¥ inf S1 _ inf S2. Let u1 � inf S1, u2 � inf S2

and suppose u1 is an upper bound of tu1, u2u. Then φpu1q ¥ φpuiq ¥ wi and hence

φpu1q ¥ w1 _ w2. Thus, u1 P R and hence inf R ¤ u1. Thus, inf R � inf S1 _ inf S2, or

equivalently, φ^pw1 _w2q � φ^pw1q _φ

^pw2q. l

Example. The condition that M is finitely generated can not be done without, unless

one imposes completeness on the induced semi-lattice. Set Q � QY t8u under the

tropical semiring structure and consider the Q-module M of functions f : N Ñ Qthat have finite support tn P N | f pnq � 8u. The map φ : M Ñ Q given by φ :

f ÞÑÀ

nPN f pnq � minnPN f pnq is well defined and a homomorphism of Q-modules.

Moreover, Q is residuated, with px : yq � x � y for y � 8. However, φ is not

residuated as a map, since t f P M | φp f q ¥ xu, for x �8 is not an order filter in Mbecause of the finiteness condition on the support.

To see that the residuatedness of N is a necessary premise of the theorem consider

the canonical injection ι : Q ãÑ R, where again R � RY t8u under tropical opera-

tions. Then the set tx P Q | ιpxq ¥ vu � tx P Q | x ¥ vu has no infimum for v P RzQ.


Example. It is not surprising, although worthwhile to point out that the dual residual

map of a homomorphism is in general not a homomorphism. We embed B � R8.

Consider the case of the submodule of B4 generated by

g1 � p0,0,8,8q, g2 � p8,8, 0, 0q, g3 � p0,8,8, 0q.

The map B3 Ñ B4 defined by px1, x2, x3q ÞÑ x1 � g1 � x2 � g2 � x3 � g3 satisfies the

conditions of Theorem 3.1.8 and hence has a dual residual map φ^. This map is not

a homomorphism of modules, since φ^pg1q � p0,8,8q, φ^pg2q � p8, 0,8q and

φ^p0,0, 0,0q � φ^pg1 � g2q � p0, 0,0q.

The final statement of this section is a distillation of the most important points

when one specializes to a certain subclass of modules over idempotent semirings with

lattice order. The proof is a direct consequence of a number of theorems in this section,

most importantly Theorem 3.1.8.

Theorem 3.1.9 Let R be a residuated semiring whose natural order is a lattice order.Consider the category ModR of modules over R that are finitely generated and embedinto a free module of finite rank. Then the following statements hold.

(i) all objects of ModR are idempotent modules whose induced order is a lattice order,(ii) the class of residuated maps and module homomorphism between objects in ModR

coincide,(iii) the dual residuation operator on morphisms maps module homomorphisms to join

semi-lattice preserving maps that respect the action of R.

Remark. The objects in the category C are essentially row and column spaces of

matrices over R. As such, examples of such modules are easy to construct. We name a

few which are of specific importance to us.

(i) (affine cones over) tropical polytopes (cf. [19]),

(ii) geometric lattices with a finite number of atoms.

3.2 Linear functionals on idempotent modules

We start by recalling some elementary definitions. Let R be a semiring. Our main

interest in this section is the map that assigns to each module M over R the collection

of homomorphisms M Ñ R. We denote this collection by M� and refer to it as the

space of linear functionals, or the (algebraic) dual space. Unlike [13] and [43] we do

not impose any continuity conditions on the linear functionals under consideration.

Clearly, M� is again a module over R under pointwise defined module operations, i.e.�φ�ψ

�pvq � φpvq �ψpvq and

�x �φ

�pvq � x �φpvq for φ,ψ P M� and x P R.

3.2. LINEAR FUNCTIONALS ON IDEMPOTENT MODULES 49

Let α : M Ñ N be a homomorphism of modules over R. Denote by α� : N�Ñ M�

the map that assigns the functional v ÞÑ ψpαpvqq to each ψ P N�. Then α� is an R-

module homomorphism, which we refer to as the transpose of α. Moreover, pα�βq� �

β� � α�, which shows that transpose and algebraic dual make up a contravariant

functor of the category of modules into itself.

The next lemma states an important property of the transpose map when we spe-

cialize to the category of modules over an idempotent semiring with induced linear

order. The first part follows by general principles, while the second part is more spe-

cific to the idempotent semirings with induced linear order. An equivalent statement

for modules over more general idempotent semirings seems not to be known (in gen-

eral injectivity of free objects in the category of modules over a fixed semiring is subtle

[69]). It is equivalent to the statement that R is injective in the category of modules

over R, and the statement that every linear functional on a submodule N � M is the

restriction of a linear functional on M . This is the form in which we require it in the

chapter on tropical linear spaces.

Lemma 3.2.1 Let α : M Ñ N be a homomorphism of modules over an idempotentsemiring R. The following statements hold.

(i) if α is surjective, then α� is injective,(ii) if R has linear canonical order and α is injective, then α� is surjective.

Proof. The first statement is an easy computation.

For the second statement, denote by IncpX , Rq the idempotent R-module of in-

creasing maps from X to R. Define the map ψ : IncpM , Rq Ñ IncpN , Rq by

�ψ f qpwq � supt f pvq | v P M and αpvq ¤ wu

We show that ψ maps M� to N�. Let f P M�. For each w P N write Sw for the

set of all u P M such that φpuq ¤ w. Then�ψ f�pwq � sup f pSwq and the linear-

ity of ψ f is equivalent to sup f pSv�wq � sup f pSvq � sup f pSwq. Clearly, Sv�w �

Sv X Sw and hence f pSv�wq � f pSvq X f pSwq. Thus, sup f pSv�wq is a lower bound

of tsupp f pSvq, supp f pSwqu and hence sup f pSv�wq ¤ sup f pSvq � sup f pSwq.

For the other inequality, let z ¤ tsup f pSvq, sup f pSwqu. It is sufficient to show that

sup f pSv�wq ¥ z. We start by showing that f pSvq is a lower set. Suppose without loss

of generality that f is surjective. Otherwise, f is the map that is identically 8 and

the image of this map under ψ is surely linear. Let x P f pSvq and suppose that y P Rsatisfies y ¤ x . There exists ux P Sv and uy P M such that x � f puxq and y � f puyq.

Then, y � x � y � f puxq� f puyq � f pux �uyq. Since ux �uy ¤ ux and Sv is a lower

set ux � uy P Sv and hence f pSvq is a lower set.

Now, since the canonical order on R is linear, there exists some x P f pSvq such

that z ¤ x . If this were not the case, then (by linearity), z would be an upper bound


of f pSvq, and hence z � sup f pSvq. Thus, there is uv P Sv and uw P Sw such that

z � f puvq � f puwq. Thus, z � f puv � uwq P f pSv�wq. Thus, sup f pSv�wq ¥ z. l

Remark. The second statement is also true in case R is an idempotent semifield.

One needs the result of [69] that the minimal injective hull of a module over an

idempotent semiring equals the maximal essential extension. Now, suppose that ι :

R ãÑ M is an essential extension, i.e. for all submodules H � M the condition ιpRq XH � t0u implies that H � t0u. Suppose for the sake of contradiction that MzιpRq �H.

Let v P MzιpRq and set H � tx � v | x P Ru. Then v P H and hence H � t0u and thus

HXιpRq � t0u. There exists some x P R such that x �v P ιpRq. However, x�1 �x �v � v PιpRq, which contradicts the choice of v. Thus, R is its own maximal extension in the

category of R-modules, and hence injective in that category. The latter is equivalent

to the statement that the pullback α� of an injection α is a surjective map.

The aim of the next part is to state a theorem that relates the algebraic dual to the

module itself, under certain conditions on the module. In passing we touch upon the

natural notion of separatedness of a module, which is naturally fulfilled for modules

over idempotent semiring that have a specific type of involution, defined in terms of

dual residues .

The map that assigns to each v P M the functional φ ÞÑ φpvq is a homomor-

phism of M to pM�q�. If this homomorphism is injective, the module M is said to

be separated by M�, or simply separated. Separatedness is equivalent to the state-

ment that for each distinct v, w P M there exists a linear functional φ P M� such

that φpvq � φpwq. In [13] the same property is also expressed by the statement that

pM , M�q is a dual pair under the canonical bracket. Our exposition here follows theirs

rather closely, save that we substitute dual residuatedness for completeness and do

not require our lattice homomorphisms to preserve arbitrary infima.

Let R be a dually residuated idempotent semiring and z0 P R. Then R is said to

be reflexive with respect to z0 if the map x ÞÑ pz0 : xq is an set-theoretic involution

R Ñ R. The notion of reflexivity as defined here is the commutative analogue of the

notion in [13]. There is a difference of convention, since their definition of natural

order is opposite to ours. Reflexivity implies in particular that the map x ÞÑ pz0 : xq is

injective and that pz0 : 0q � infty | z0 ¤ 0u � inf R is defined. Note that this infimum

does not exist in tropical semifields other than the Boolean semifield. Finally, a dually

residuated idempotent semiring is said to be reflexive if it is reflexive with respect to

at least one of its elements.

The relation between both concepts defined above is that a module M over a

reflexive semiring R is separated when that module is dually residuated over R. Fix z0

such that R is reflexive with respect to z0. The vehicle for the proof is a natural map

M Ñ M� defined by v ÞÑ inftφ | φpvq ¥ z0u. We write φv for the image of v under

3.2. LINEAR FUNCTIONALS ON IDEMPOTENT MODULES 51

this map. The following statement shows that this map exists and gives an explicit

form. Separatedness is an immediate corollary.

Lemma 3.2.2 Let R be an idempotent semiring that is reflexive with respect to z0 andlet M be a module over R that is dually residuated over R. Then the map M Ñ M� givenby v ÞÑ φv is well-defined and equals w ÞÑ pz0 : pv : wqq. In particular, M is separatedby M�.

Proof. See Theorem 36 of [13] and its corollary. l

Remark. Common operations on semirings preserve reflexiveness (cf. Proposition 31

[13]). The example that is of most interest us is the following. Let F be a semifield

whose induced order is linear. Denote the minimal lattice that has a minimal element

and contains F as a sublattice by F . The underlying set of F is equal to the underlying

set of F (if and only if F is finite), or equal to the disjoint union of the underlying set

of F and tKu, where K is the minimal element of F . The multiplication operator on Fextends to F by setting K� x � x � x �K � x for all x P F and K�K � K. We refer to Fas the extended semifield of F , by abuse of terminology, since if F is infinite K has no

multiplicative inverse and F is only a semiring, since K has no multiplicative inverse.

For modules over idempotent semirings it is not so that the module is isomor-

phic to its algebraic dual as a module, even under strong contextual conditions. The

theorem is somewhat more subtle. It is nicely stated in terms of the notion of an oppo-

site module, which we define in the setting of finitely generated modules over lattice

ordered semirings that are residuated over that semiring.

Let R be a semiring whose induced order is a lattice and let M be a finitely gener-

ated module over R. Then, by Proposition 3.1.6, it follows that M has induced lattice

order. The opposite module of M is the module with underlying set that of M , addi-

tion defined by v�1w � v_w, and scalar operation given by x �1 v � inftw | x �w ¤ vu.It is of course not automatic that this exists for every x P R and v P M . If it does the

opposite module is well-defined and we denote it by M op.

Theorem 3.2.3 Let R be an idempotent semiring that is reflexive with respect to z0 andlet M be a module over R that is both dually residuated and finitely generated over R.Then M and M� are anti-isomorphic lattices with isomorphisms given by v ÞÑ inftφ |

φpvq ¥ z0u and φ ÞÑ inftv | φpvq ¥ z0u. In other words, these maps are isomorphismsbetween the additive semigroups of M and pM�qop.

Proof. This is essentially Corollary 33 of [13]. Although completeness is required in

their statement, examination of the proof reveals that in fact only certain dual residua

are required to exist. Such dual residue exist in M by the premise, and they exist in

M� since M� embeds into a free module of finite rank by the premise that M is finitely


generated. That dual residua also exist in pM�qop is now a consequence of statement

(18b) of the article. l

Example. Certain natural submodules of the algebraic dual are infinitely generated,

even if the algebraic dual is finitely generated. The key property is the following. A

semiring is called zero-sum free when it holds that x � y � 0 implies both x � 0

and y � 0. Idempotent semirings have this property, since x � y � 0 implies that

x � x � 0 � x � x � y � x � y � 0, and similarly for y . Let R be a zero-sum free

semiring and M a module over R. Consider the space of linear functionals M� of Minto R. Let v P M . It is surprising that the set tφ P M� | φpvq � 8u Y t0M�u is a

submodule of M�. After all, φpvq �ψpvq � 8 when both φpvq � 8 and ψpvq � 8.

However, this submodule need not be finitely generated, even if M� is.

As a specific example, consider the module M � Q28 over the tropical rational

numbers and let v � p0,8q. Set M�v � tφ P M� | φpvq � 8u and write εi for the

i-th coordinate function on Q28. Then M�

v consists of the tropical linear combinations

px1 d ε1q ` px2 d ε2q having either x1 � 8 or x1 � x2 � 8. This module is not

finitely generated. The easiest way to see this is by Theorem 3.1.8, since (i) the map

φ ÞÑ φpvq is not dually residuated, since the infimum of tφ P M�v | x1 � φpvq ¥ 0u

does not exist, (ii) the image of M�v zt8u is contained in Q8zt8u and (iii) Q8 is

residuated.

3.3 Ranks and vanishing conditions

This section deals with putting the algebraic theory of idempotent semirings and rings

on equal footing. The main item is the so-called tropical vanishing property of finite

sequences, which specializes both to the algebraic definition of a vanishing sum and to

the notion of vanishing of tropical geometry. This section was inspired by the article

[17] and the different notions of dependence in tropical modules that exist in the

literature. See [1] for an overview.

We require the next definition.

Definition 3.3.1 Let R be a semiring. The indicator εR of R is the nullary operator on

R that equals �1R if it exists in R, and 1R otherwise. Note that εR ��1R if and only if

R is a ring.

The indicator essentially serves to distinguish rings from proper semirings (i.e.

those in which the additive monoid is not a group) in a way that is useful in identities

and definitions involving the algebra of the semiring.

There exist several notions of dependence in a module in the literature that are of

interest to us: (i) weak dependence, (ii) dependence in the sense of Gondran-Minoux,

3.3. RANKS AND VANISHING CONDITIONS 53

and (iii) tropical dependence. All of these notions are well-studied. For an overview,

see [1]. We postpone their definition until the end of this section.

The different notions of dependence are related to different notions of vanishing

in a module. The one corresponding to the second type of dependence is a straightfor-

ward translation. Let M be a module over a semiring R and let pv1, . . . , vnq be a finite

sequence of elements of M . This sequence is said to vanish in the sense of Gondran-Minoux if there exists a partition pP,Qq of rns such that,

¸iPP

vi � εR

¸iPQ

vi .

The partition is allowed to be trivial. If one of the sets in the partition can be chosen

a singleton, then the sequence is said to vanish weakly. Clearly, a finite sequence that

vanishes weakly also vanishes in the sense of Gondran-Minoux.

The sequence pv1, . . . , vnq is said vanish in the tropical sense when for each φ P M�

there exists a partition pPφ ,Qφq of rns such that

¸iPPφ

φpwiq � εR

¸iPQφ

φpwiq.

This notion is useful for modules over semirings with induced linear order whose

algebraic duals are finitely generated, because in that case, the potentially infinite

number of condition (one for each linear functional) reduces to a finite number (one

for each generator of the algebraic dual).

The algebraic dual of the module R consists of the maps x ÞÑ a�x , where a P R, and

hence tropical and Gondran-Minoux vanishing are equivalent in R. For sequences in Rwe simply say they vanish, or refer to them as vanishing sequences, when the vanish in

one of either equivalent manners. One could say that a sequence in a module vanishes

tropically if its image under any linear functional on the module vanishes. In general,

it is the case that any sequence that vanishes in the sense of Gondran-Minoux also

vanishes in the tropical sense. The converse, however, is false. In the case of modules

over rings, however, all notions of vanishing are equivalent. We summarize matters in

the next statement.

Proposition 3.3.1 Let M be a module over a semiring R.

(i) Any finite sequence in M that vanishes weakly, vanishes in the sense of Gondran-Minoux.

(ii) Any finite sequence in M that vanishes in the sense of Gondran-Minoux vanishestropically.

When M is the module R, then the implication of (ii) is an equivalence. If R is a ringand M is separated by M�, then both implications (i) and (ii) are equivalences, and a


sequence pv1, . . . , vnq vanishes if and only if°n

i�1 vi � 0.

Proof. Statement (i) is immediate. Statement (ii) is a consequence of the linearity of

the elements of M�.

Suppose R is a ring and M is separated by M�. If a finite sequence indexed by

rns vanishes in the sense of Gondran-Minoux, then any partition of rns is a vanishing

partition. Thus, it vanishes weakly. If a sequence v1, . . . , vn vanishes tropically, then

for any partition pP,Qq of rns and φ P M�,

¸iPP

φpviq � �¸jPQ

φpv jq.

Hence φp°n

i�1 viq � 0. Thus, by separatedness,°n

i�1 vi � 0. l

Examples of modules over semiring in which tropical vanishing is not equivalent

to Gondran-Minoux vanishing are well-known, and we give one later. The notions of

vanishing lead to two distinct notions of structure preserving maps, as defined next.

Let M be an R-module and N an S-module for semirings R and S. A map α : M Ñ

N is said to be a Gondran-Minoux pseudomorphism of modules when for all v, w P Mthe following conditions are satisfied,

(i) αp0M q � 0N ,

(ii) αpεR � vq � εS �αpvq,(iii) the sequence pαpvq,αpwq,εS � αpv � wqq vanishes in the sense of Gondran-

Minoux.

If instead of (iii) the sequence pαpvq,αpwq,εS �αpv�wqq vanishes in the tropical sense,

then the map α is said to be a tropical pseudomorphism of modules. The terminology

extends to maps between semirings and algebras over semirings. Such maps are said

to be pseudomorphisms of semirings, or pseudomorphisms of algebras, if they preserve

the multiplicative structure, in addition to the above axioms.

Remark. Denote the ring of scalars of M by R and the ring of scalars of N by S.

Clearly, if S � R, and α is a homomorphism of R-modules it is a pseudomorphism.

Moreover, if N � S, then by Proposition 3.3.1, a map α : M Ñ N is a Gondran-

Minoux pseudomorphism if and only if it is a tropical pseudomorphism, and we omit

the classifier.

Example. Let K be a field equipped with a non-Archimedean valuation ω : K Ñ Fthat is surjective onto a tropical semifield F � R8. Note that we use the term valua-

tion in the sense of [5]. The more descriptive term would be additive ring valuation.

We denote the operators on F by ` and d, as is customary for tropical semifields.

Then, ω is a pseudomorphism of semirings. The valuation ω respects the multiplica-

tive structure, since ωpx yq � ωpxq �ωpyq � ωpxq dωpyq, where � denotes the


(extension of the) ordinary addition operator on R. Pseudomorphism axioms (i) and

(ii) state that ωp0q � 8 and ωp�1q � 0, which are fulfilled. The vanishing axiom

(iii) states that either,ωpx� yq �ωpxqdωpyq �mintωpxq,ωpyqu, orωpxq �ωpyqand ωpx � yq ¥ωpxq. This property is satisfied by ω.

The importance of Gondran-Minoux pseudomorphisms lies in the following propo-

sition. Note that we refer to axiom (iii) and its tropical counterpart as the vanishingaxiom.

Proposition 3.3.2 Let M be a module over R and N a module over S. Consider anarbitrary map α : M Ñ N that satisfies αp0M q � 0N and αpεR � vq � εS � αpvq forall v P M. Then α is a Gondran-Minoux pseudomorphism if and only if for each se-quence pv1, v2, . . . , vnq P M n that vanishes in the sense of Gondran-Minoux, the imagepαpv1q,αpv2q, . . . ,αpvnqq vanishes in the sense of Gondran-Minoux as well.

Proof. We start by proving the reverse implication. Let v, w P M . The se-

quence pv, w,εRpv � wqq clearly vanishes. Thus, by the premise, the sequence

αpvq,αpwq,αpεRpv�wqq � pαpvq,αpwq,εSαpv�wqq vanishes.

Let pP,Qq be a vanishing partition for v1, . . . , vn. In particular, there exists mutually

disjoint sets P1, . . . , Pm and Q1, . . . ,Qn that together partition rns such that

m

j�1

α�¸

iPPj

vi

�� εS

n

j�1

α� ¸

iPQ j

vi

�.

Choose these sets such that maxt|Pi |, . . . , |Pm|, |Q1|, . . . , |Qn|u is minimal and suppose

that the maximum is larger than one and that the number of sets of maximal size

is minimal. Without loss of generality, since ε2S � 1, let |Q1| attain the maximum and

write w1, . . . , wk for the elements indexed by Q1. By the vanishing condition εSαpw1�

. . .�wkq,αpw1q,αpw2 � . . .�wkq vanishes. Thus, at least one of the following holds,

(i) αpw1 � . . .�wkq � αpw2 � . . .�wkq �αpw1q,

(ii) εSαpw2, . . . , wkq � εSαpw1 � . . .�wkq �αpw1q,

(iii) εSαpw1q � εSαpw1 � . . .�wkq �αpw2 � . . .�wkq.

We show that all cases lead to a contradiction.

(i) Immediate.

(ii) Adding αpw1q to both sides gives a contradiction.

(iii) Adding αpw2 � . . .�wkq to both sides gives a contradiction.

Hence, the maximal size is 1 and the expression αpv1q � . . .�αpvnq vanishes. l

Example. A potentially confusing issue is that the axioms defining a Gondran-Minoux

pseudomorphism are not preserved under common operations on maps. For example,

let K be a field and ω : K Ñ B a surjective valuation onto the trivial semifield B �


t0,8u. Then ω is a Gondran-Minoux pseudomorphism between K and B. However,

the coefficient-wise application ofω is not a Gondran-Minoux pseudomorphism K3 Ñ

B3. Set v � pa, b, 0q and w � p�a, 0, cq, with a, b, c P K�. The images of v, w and

v � w in B3 are p0, 0,8q, p0,8, 0q and p8, 0, 0q, whose sum does not vanish in the

sense of Gondran-Minoux. Note however that it does vanish tropically and that the

coefficient-wise application of ω is a tropical pseudomorphism.

According to Proposition 3.3.2 above the Gondran-Minoux pseudomorphisms

have the desirable property that they form the largest class of maps that respect the

notion of Gondran-Minoux vanishing in the entire category of modules over a semi-

ring. However, by the example, such maps are not suited for the purpose of tropical

geometry, since the coefficient-wise application of an additive valuation is not in the

class. The solution to this dilemma is to restrict to a full subcategory of modules over

a ring, in which the strictly larger class of tropical pseudomorphisms shares some of

the desirable properties of that of the Gondran-Minoux pseudomorphisms. The next

lemma is essential in proving that this works.

Lemma 3.3.3 Let α : M Ñ N be a tropical pseudomorphism and suppose N is separatedby N�. If any of the following conditions hold,

(i) N is a module over a ring,(ii) both M and N are modules over semirings with idempotent addition,

then α is an additive map. Moreover, if (i) holds and 2 � imα � t0Nu, then M is also amodule over a ring.

Proof. We prove the claims in the order stated. Let v, w P M .

(i) For each φ P N� the sum �φpαpv � wqq �φpαpvqq �φpαpwqq vanishes. This

implies that φpαpv � wqq � φpαpvq � αpwqq, by linearity of φ. Since the canonical

morphism N Ñ N�� is injective it follows that αpv�wq � αpvq �αpwq.(ii) Let φ P N�. Denote the composition φ � α by ψ. Then ψ : M Ñ S is a

pseudomorphism, where S is the semiring of scalars of N . By the arguments in the

previous item, it is sufficient to prove that ψpv � wq � ψpvq �ψpwq. By the pseudo-

morphism axioms, the sumψpv�wq�ψpvq�ψpwq vanishes. This implies that either

ψpv � wq � ψpvq �ψpwq, ψpvq � ψpv � wq �ψpwq, or ψpwq � ψpv � wq �ψpvq.By symmetry, it is sufficient to consider the case ψpvq � ψpv � wq �ψpwq. By idem-

potency of the addition in S the equality ψpvq�ψpwq �ψpv�wq�ψpwq and hence

ψpv�wq ¥ψpvq �ψpwq.For the other inequality, consider the image of v � w � w under ψ. By the pseu-

domorphism axioms, the sum ψpv � w � wq � ψpv � wq � ψpwq vanishes. Thus,

ψpwq � ψpv � wq, or ψpv � wq � ψpwq � ψpv � wq. In terms of the natural or-

der, ψpv � wq ¤ ψpwq. The inequality, ψpv � wq ¤ ψpvq follows by symmetry. Thus,

ψpv�wq ¤ψpvq �ψpwq, and the statement follows.


Denote the semiring of scalars of M by R. Let v P M such that αpvq � αpvq � 0.

Suppose that R is not a ring. Then εR � 1 and hence αpvq � �αpvq. This contradicts

the choice of v, and hence R is a ring. l

The different notions of vanishing lead to different notions of rank as well. Con-

sider a finite sequence v1, . . . , vn of elements of M . Such a sequence is said to be

Gondran-Minoux, tropically, or weakly dependent if there exist scalars x1, . . . , xn P R,

not all equal to 0, such that the sequence x1 � v1, . . . , xn � vn vanishes in the sense of

Gondran-Minoux, tropically, or weakly, respectively. It is said to be independent if it is

not dependent. If S � M is some subset, then the Gondran-Minoux, tropical, or weak

rank of this set is the maximal size of a subset of V that is independent in the sense of

Gondran-Minoux, tropically, or weakly. For a more complete overview of the different

notions of rank we refer to reader to [1]. An interesting paper on the notion of weak

rank in specific classes of modules over idempotent semirings is [68]. Note that what

we refer to as weak dependence is referred to as dependence in that paper.

For us the notion of tropical rank is of most importance, since it is most strongly

connected to tropical linear spaces. Although the tropical rank of a matrix was already

defined and studied in the paper [18] it was first connected to a notion of dependence

in [35], albeit in the context of supertropical algebra. Tropical rank is most useful for

modules whose semiring of scalars has linear induced order and we postpone further

discussion of tropical rank to the end of the next section.

The weak rank of a set of vectors is most important when restricted to subsets

of modules over semifields. In that case it is strongly connected to the generating di-

mension of the module, i.e. the minimal cardinality of a spanning subset. The theorem

relating generating dimension to weak dependence it best stated in terms of a concept

we borrow from lattice theory. Let M be a module over an idempotent semifield F and

u P N � M . Then u is said to be reducible in M if there exist distinct v, w P Mztuusuch that u� v�w, and it is said to be irreducible otherwise. The following statement

and their proofs can be found in [68] and are quite well known for finitely generated

modules over the tropical real numbers (cf. [19, 27]). We reformulate the relevant

statements in [68].

Theorem 3.3.4 Let S � M be a finite set with M a module over an Archimedean idem-potent semifield. Then S is weakly independent if and only if every u P S is irreducible inthe module generated by S.

Proof. This is Proposition 2.5.3 of [68]. l

A basis of a module M is a weakly independent generating set for M . The previous

theorem has the next statement as an immediate corollary.

Corollary 3.3.5 Let M be a finitely generated module over an Archimedean idempotent


semifield. Then it has a unique basis up to scalar multiples.

Proof. This is Theorem 5 of [68]. l

3.4 Modules over linearly ordered semirings

The next lemma is particular for the case of modules with a linear order, and states

an essential non-trivial property that holds for such modules.

Lemma 3.4.1 Let M be a module with linear canonical order. Let k and n be naturalnumbers. For each i P t1, . . . , ku, let σi be a Gondran-Minoux vanishing sequence oflength n with elements in M. Then, the sequence σ1 � . . .� σk is a Gondran-Minouxvanishing sequence as well.

Proof. Set τ� σ1�. . .�σk and let j P t1, . . . , nu be the index of a minimal element of

τ. There exists i such that j is the index of a minimal element of σi . By the character-

ization of vanishing sequences in modules with induced linear order there is another

index k P t1, . . . , nu such that pσiq j � pσiqk. Thus, τ j ¤ τk ¤ pσiqk � pσiq j � τ j and

τ j � τk are both minimal entries of τ. l

The above statement is the main reason to restrict to modules over semirings Rhaving linear induced order. It states equivalently that the collection of vanishing

sequences of fixed length n with elements in R is a module over R. The implication

is that the analogue of linear tropical prevarieties over a semiring are modules over

this semiring and hence that they behave much like actual tropical prevarieties and

tropical convex sets (cf. [19]). In particular, the statement is an essential ingredient

in the proof that the tropical linear spaces of Chapter 4 are both finitely generated

modules and tropical prevarieties.

Remark. The condition that M has linear canonical order seems to be necessary in the

above statement. The essential obstruction to proving the statement for semiring with

non-linear canonical order seems to be that the vanishing partitions of the different

sequences pσiq may differ. If this were not the case, then the sequence τ in M k of

length n that is defined by τ j � ppσ1q j , . . . , pσkq jq is a vanishing sequence. Since the

map M k Ñ M given by pv1, . . . , vkq ÞÑ v1 � . . .� vk is a Gondran-Minoux pseudomor-

phism the entry-wise image of τ under this map would vanish as well, by Proposition

3.3.2. This image is precisely the sum σ1 � . . .�σk.

Example. This an example of Gondran-Minoux vanishing sequences whose sum is not

a Gondran-Minoux vanishing sequence, when the induced order on M is not linear.

3.4. MODULES OVER LINEARLY ORDERED SEMIRINGS 59

Let M be the module B4. Define the Gondran-Minoux vanishing sequences σ and

τ by,

σ ��p8,8, 0, 0q, p8,8,8, 0q, p8,8, 0,8q

�τ �

�p0,8,8,8q, p0, 0,8,8q, p8, 0,8,8q

�.

The sum σ�τ is the sequence,

σ�τ��p0,8, 0, 0q, p0, 0,8, 0q, p8, 0, 0,8q

�.

Thus, σ�τ is not a Gondran-Minoux vanishing sequence. This shows that the condi-

tion that the canonical order on M is linear is not superfluous.

Remark. There remains the question whether there is a natural class of modules on

which the implication of Lemma 3.4.1 is an equivalence. A natural such condition

might be stated in the following way. Let M be a module with idempotent addition

and denote by k the minimal number such that for every v1, . . . , vn P M there exists

a subset S � rns of size k such that°n

i�1 vi �°

jPS v j . For example, M has linear

induced order precisely when k � 1.

The above example shows that any module into which B4 embeds has vanishing

sequences whose sum is not a vanishing sequence. Modules over idempotent semir-

ings which have k ¥ 2 always contain a B-submodule isomorphic to B2. Hence, to

prove equivalence in Lemma 3.4.1 it would be sufficient to find two vanishing se-

quence of equal length, whose elements are in B2 and whose sum does not vanish.

The next corollary is important since it shows that for a sequence to tropically

vanish, one needs only to check vanishing of the image sequence under a finite num-

ber of linear functionals, if the algebraic dual is finitely generated. In particular, any

module over a ring whose induced order is linear that is a submodule of a finitely

generated free module has a finitely generated algebraic dual, by Lemma 3.2.1. In

particular, tropical convex subsets of Rn8 and tropical linear spaces are of this type.

Corollary 3.4.2 Let R be a semiring with linear order and M a module over R whosealgebraic dual is generated by some finite set Γ. A finite sequence pv1, v2, . . . , vkq vanishestropically if and only if for each γ P Γ the sequence pγpv1q,γpv2q, . . . ,γpvkqq vanishes inR.

Proof. Let φ P M�. We show that pφpviqq vanishes. Write φ �°γPΓµγ � γ with

µγ P R. The sequence pφpviqq is the sum of the sequences µγ � pγpviqq, all of which are

vanishing sequences in R. By the lemma, pφpviqq is a vanishing sequence R. l


The corollary states in particular that a finite subset tv1, . . . , vku � Rn is tropically

dependent if there exist α1, . . . ,αk P R such that the minimum in the expression α1 �

v1i � . . .� αk � vki is attained at least twice for i P rns, where vi j is the j-th entry of

the vector vi . Thus, for free modules over linearly ordered semirings tropical rank is

the same as the notion in [35]. Moreover, our notion of vanishing corresponds to the

notion of summing to a ghost element. Hence, we have the following theorem.

Theorem 3.4.3 Let V � Rn be a finite set of vectors. The tropical rank of V is the sizeof the largest non-vanishing minor of the matrix whose rows are the elements of V .

Proof. This is Corollary 3.14. of [35]. l

3.5 Conclusion

Tropical linear spaces over R are in their most general linear prevarieties in a free

module over R of finite rank. The aim of this chapter was to restrict the choice of Rsuch as to have a comfortable setting in which to think about tropical linear spaces. A

number of properties are convenient to have.

(i) Duality: tropical linear spaces are finitely generated modules.

(ii) Tropical rank: there is a meaningful notion of tropical rank.

(iii) Residuation: homomorphisms between tropical linear spaces are residuated

mappings.

(iv) Lattice: the induced order on a tropical linear spaces is a lattice order.

(v) Unique basis: tropical linear spaces (as modules) have an essentially unique

basis.

All of these conditions are fulfilled if R is a tropical semifield. If R is a residuated idem-

potent semifield whose induced order is linear (or if R is an idempotent semifield with

induced linear order) then all conditions except possibly the last one are fulfilled.

Chapter 4

A category of tropical linear

spaces

Inspired by Figure 4.1(a) from talks by Mikhalkin (see [39, Paragraph 12]), in which

three points on a tropical projective line are brought into special position by a se-

quence of modifications and projections (except the last map, which is an invertible

translation corresponding to an invertible diagonal tropical matrix), we aim to define

and study tropical combinatorial analogues of affine algebraic groups. A first attempt

would be to define these as certain groups of invertible tropical matrices. However, it

is easy to see that all invertible tropical matrices are in fact monomial, and the process

in Figure 4.1(a) cannot be captured by a single monomial matrix (though the last step

can).

(a) Moving three points to standard position onthe projective line.

(b) A morphism from thetropical projective line toitself.

Figure 4.1: Moving three points, stepwise or at once.

The composition of the morphisms in Figure 4.1(a) is depicted in Figure 4.1(b):

it is a metric tree with two isometric embeddings of the tropical projective line

61

62 CHAPTER 4. A CATEGORY OF TROPICAL LINEAR SPACES

1 23

a b ∞c d ∞∞ ∞ a� d

a� d < b� c

56

a b ∞c d ∞∞ ∞ b� c

a� d > b� c

4

89

a b ∞c d ∞∞ ∞ e

a� d = c� b < e

a ∞ ∞∞ d ∞∞ ∞ a� d

∞ b ∞c ∞ ∞∞ ∞ b� c

7 10

a b ∞c d ∞∞ ∞ e

a� d = b� c = e

a ∞ ∞b d ∞∞ ∞ a� d

a b ∞∞ d ∞∞ ∞ a� d

∞ b ∞c d ∞∞ ∞ b� c

a b ∞c ∞ ∞∞ ∞ b� c

12

3

4

5

6

78

9

10

Figure 4.2: TPGL2: polyhedral complex and matrices. The dashed lines and the dashed(white) vertices are not really there, so that, for instance, face 3 has the topology of ahalf-plane.

r�8,8s—one for the source (solid) and one for the target (dashed). Moreover, the

tree does not have ends that are superfluous in the sense that they are covered by

neither the source nor the target. Such pairs of embeddings form a pure polyhedral

complex of dimension three (see Figure 4) which is a monoid with a natural composi-

tion: embed everything into a larger tree, and delete superfluous ends. Therefore, it is

reasonable to call this complex TPGL2, the tropical projective general linear group of

the line. Incidentally, it is isomorphic, as a polyhedral complex with monoid structure,

to the tropicalization of the variety GL2 as sitting inside 3� 3-matrices in the usual

way (modulo scalars, to be precise). In higher dimensions, we still have a monoid

homomorphism from TGLn to the tropicalization of GLn, but it is not injective.

Our aim for the remainder of this section is to give an appropriate construction

that mimics Figures 4.1 and 4.2 for tropical linear spaces other than R28. A clue on

how this is to be done is given by the observation that generic morphisms from the

tropical projective line to itself consist of a pair of embeddings into a three-valent trees

on four leaves. The latter are common objects in the tropical geometry literature. They

are projectivizations of generic 2-dimensional tropical subspaces of R48.

Our approach is to define a category, whose objects include Rn8, and define the

tropical linear group of rank n to be some subgroup, or submonoid, of the endomor-

phisms of Rn8 in this specific category. This is the program that we attempt to carry

out in this chapter.

4.1. VALUATED MATROIDS AND TROPICAL LINEAR SPACES 63

The category is constructed in terms of general tropical linear spaces, of which Rn8

is a specific instance. Tropical linear spaces are modules with a combinatorial struc-

ture that abstract the properties of tropicalizations of vector spaces over valued fields,

in the same way that matroids abstract properties of linear independence. They were

implicitly introduced in [23] and put into a tropical context in [59]. It is inconvenient

that tropical linear spaces have no intrinsic definition. The definition of a tropical lin-

ear space is through an intermediate object called a valuated matroid. Section 4.1 is

devoted to gathering and reformulating known results on tropical linear spaces and

valuated matroids. A difference between our approach and the more common one in

the tropical geometry literature is that we tend to understand tropical linear spaces as

finitely generated submodules, instead of as tropical prevarieties, and that we develop

the theory over arbitrary tropical semifields. The latter does not yield new results, but

makes the connection to the theory of matroids cleaner.

A morphism between tropical linear spaces L andM is defined to be a tropical

linear subspace of L �M that is obtainable from L by iteratively taking graphs oflinear functions. In Figure 4.1(b) the morphism is the tree with four leaves in the

middle. It is a twofold linear extension of the tropical projective line, with each linear

extension glueing a branch onto the projective line. The concept of a general linear

extension is developed in Section 4.2. The actual category and TGLn are defined in

Section 4.3.

4.1 Valuated matroids and tropical linear spaces

This chapter serves to gather results on tropical linear spaces and valuated matroids

from various sources, and to restate and prove them in a unified language. All of the

results in the chapter are known in some way.

In Section 4.1.1 we start by defining valuated matroids. These are best seen as

functions that assign a finite element of a tropical semifield to each basis of a matroid.

They include the concept of an ordinary matroid by taking t0,8u as the tropical semi-

field. To each valuated matroid we associate a finitely generated semimodule called a

tropical linear space. This is not the ordinary definition of a tropical linear space seen

for example in [59], but is more convenient for our purposes. It is also more fitting

with the algebraic approach to tropical linear spaces we adopt here. The remainder of

Section 4.1.1 is devoted to stating and proving some important properties of tropical

linear spaces and valuated matroids.

In Section 4.1.2 we give constructions to make valuated matroids from other ones.

The main purpose is to understand which operations on tropical linear spaces yield

tropical linear spaces, and which operations are induced on the valuated matroids to

which the tropical linear spaces are associated. For example, the Cartesian product of


two tropical linear is a tropical linear space, as is the coordinate projection.

In the last short Section 4.1.3 we deal with parameter spaces of tropical linear

spaces.

4.1.1 Foundations

Valuated matroids

The main objects under consideration are introduced in [23] under the name valuatedmatroids. They were put into a tropical context in [59]. To emphasize the connection

with matroids and the dependence on the tropical semifield F we use the names

matroid over F , F-valuated matroid, or even F-matroid, for shortness.

Definition 4.1.1 Let R be a finite set, F a tropical semifield and d P N. A valuatedmatroid on R over F of rank d is a function µ : PdpRq Ñ F that satisfies the valuatedexchange condition for all X , Y P PdpRq, i.e. for all x P X zY there exists y P Y zX such

that

µpX q dµpY q ¥ µpX � x � yq dµpY � x � yq. (4.1)

Valuated matroid µ and ν on R over F are called proportional if they are scalar mul-

tiples of each other, i.e. if there exists λ P F� such that µpX q � λ d νpX q for all

X � R.

Proposition 4.1.1 (Immediate properties) Let µ be a valuated matroid on R over F.(i) For h : F Ñ G a homomorphism of semifields, the composition h �µ is a valuated

matroid on R over G.(ii) The set tA� R | µpAq � 8u forms the collection of bases of an ordinary matroid.

(iii) The set tA � R | µpAq is minimalu forms the collection of bases of an ordinarymatroid.

Proof. We prove the claims in the order in which they are stated. All proofs are

elementary, which we give in full for the sake of completeness.

(i) Let A and B be subsets of R of size d. Let a P AzB. There exists b P BzA such

that µpAq d µpBq ¥ µpA� a � bq d µpB � a � bq. Since h is increasing and a

multiplicative homomorphism, hpµpAqqdhpµpBqq ¥ hpµpA� a� bqqdhpµpB�a� bqq.

(ii) Let A and B such that µpAq � 8 and µpBq � 8. Let a P AzB. There exists

b P BzA such that µpA� a� bq d µpB � a� bq � 8. Thus, µpA� a� bq � 8

and µpB� a� bq � 8, since F has no zero divisors.

(iii) Let A and B be such that both µpAq and µpBq are minimal. Then for each a P AzBthere exists b P BzA such that µpA� a� bqdµpB� a� bq ¤ µpAqdµpBq. Thus,

µpA� a� bq and µpB� a� bq are minimal.


l

The second and third item in the above proposition give natural ways of asso-

ciating ordinary matroids to valuated matroids. These are made explicit in the next

definition.

Definition 4.1.2 Let µ be a valuated matroid on R over F . The underlying ma-troid of µ is the matroid on R that has tX � R | µpX q � 8u as its collection

of bases. The residual matroid or minimal matroid is the matroid on R that has

tX � R | µ is minimal at Xu as its collection of bases. We write matpµq for the un-

derlying matroid, and respµq for the residual matroid of µ.

Proposition 4.1.2 The map µ ÞÑ matpµq establishes a bijection between valuated ma-troids over B and ordinary matroids.

Proof. This is a trivial consequence of part (ii) of Proposition 4.1.1 and the fact that

µpAq � 8 implies µpAq � 0. l

Remark. If we reinterpret matroids as valuated matroids over B, which is valid by

the above proposition, then the map µ ÞÑ matpµq is induced in the sense of part (i)

of Proposition 4.1.1 by the homomorphism F ÞÑ B that maps 8 to 8 and everything

else to 0.

The underlying matroid is used to transfer matroid terminology to valuated ma-

troids. We shall for example speak of the rank of the valuated matroid to mean the

rank of the underlying matroid. Similarly, the bases of a valuated matroid are the

bases of the underlying matroid, i.e. the collection of sets on which the valuated ma-

troid is finite.

An important class of valuated matroids is formed by the realizable valuated ma-troids. The class depends on a field K with a valuation ω : K Ñ F . Let A be a d � n-

matrix where d ¤ n and suppose that A has maximal rank. For each subset J of

t1, . . . , nu we write ArJs for the submatrix of A consisting of the columns indexed by

J . Then the function J ÞÑ ωpdetpArJsqq is a valuated matroid; the valuated matroid

associated to A. A valuated matroid realizable over K , or pK ,ωq if we wish to be pre-

cise, is a valuated matroid that arises in this way. This example hints at the relation

between tropicalized linear spaces and valuated matroids, which we discuss later.

Before moving on to tropical linear spaces, we state and prove a number of alter-

native characterizations of valuated matroids, that put them more clearly in a tropical

context, and relate them to subdivisions of matroid basis polytopes. Note that part

(iv) is essentially the definition found in [59].For brevity, we write Si j for the set SYti, ju andBpµq denotes the set of bases of

the valuated matroid, i.e. the collection of size d sets whose image under µ is finite.


Theorem 4.1.3 (Speyer, Murota) Let R be a finite set and F a tropical semifield, andµ :P pRq Ñ F a function. Then the following statements are equivalent,

(i) µ is a valuated matroid of rank d,(ii) µ satisfies the exchange condition (4.1) for all subsets A, B � R of size d such that

|AzB| � 2,(iii) the minimum of the expression,

àiPY zX

µpY � iq dµpX � iq (4.2)

is attained at least twice, for all X , Y � R satisfying |Y | � d � 1 and |X | � d � 1,(iv) the minimum of (4.2) is attained at least twice, for all X , Y � R satisfying |Y | �

d � 1, |X | � d � 1 and |Y zX | � 3,(v) for each S � R with |S| � d � 2 and i, j, k, l P RzS the minimum of�

µpSi jq dµpSklq`�µpSikq dµpS jlq

`�µpSilq dµpSk jq

is attained at least twice.(vi) the convex hull of tχB | µpBq � 8u is a matroid basis polytope in the sense of [9]

and the assignment χB ÞÑ µpBq P F� � R of values to the vertices of this polytopeinduces a subdivision all of whose faces are again matroid polytopes.

Proof. We establish this theorem by first proving the equivalence of (i) and (ii) and

the equivalence of (i) and (iii). The equivalence of (ii) and (iv) is then a byproduct of

the latter proof. Part (v) is simply a restatement of (iv), with S � X X Y . Finally, we

prove the equivalence of (iv) and (vi).

We start with the equivalence of (i) and (ii). It is trivial that (i) implies (ii). That

(ii) also implies (i) is more complicated. We follow the proof by contradiction in [48].Assume without loss of generality that F contains at least three elements. Otherwise,

extend F .

Suppose that µ satisfies the basis exchange condition for all pairs pA, Bq satisfying

|A| � |B| � d and |A∆ B| � 4 and that there exist pairs which do not satisfy the basis

exchange condition. Let pA�, B�q be such a pair, chosen such that |A� ∆ B�| ¡ 4 is

minimal. The aim is to arrive at a contradiction by constructing a pair not satisfying

the basis exchange condition whose sets have a symmetric difference that has strictly

smaller size. In fact, it is sufficient to show that this pair does not satisfy the basis

exchange condition for τ �µ, for some τ P FR�, with,

pτ �µqpBq ��ä

iPBτi

dµpBq, B PPdpRq,

since τ �µ is a valuated matroid if µ is.


Let a� P A�zB� be such that for all b P B�zA� the strict inequality µpA�qdµpB�q µpA�� a�� bqdµpB�� a�� bq holds. We choose τ P FR

� and set µ1 � τ �µ such that

µ1pA�q � µ1pA��a��iqwhen i P B�zA� and A��a��i PBpµq, and µ1pB��a��iq ¡µ1pB�q if i P B�zA�. For example, we might choose,

τi �

$'&'%µpA�q �µpA�� a�� iq, i P B�zA� and A�� a�� i PBpµq,µpB�� a�� iq �µpB�q � ε, i P B�zA� and A�� a�� i RBpµq,0 otherwise.

,

for some ε ¡ 0.

Let a0 P A�zB� such that a0 � a�. Choose b0 P B�zA� such that µ1pB� � a0 � bqis minimal for b � b0 and set B0 � B� � a0 � b0. It is sufficient to show that µ1pB0 �

a�� bq ¡ µpB0q for all b P B�zA�, since then

µ1pA�q dµ1pB0q � µ1pA�� a�q dµ1pB0q

µ1pA�� a�� bq dµ1pB0 � a�� bq,

for all b P B0zA� � B�zA� and |A� ∆ B�| ¡ |A� ∆ B0|.

Let b P B�zA� and consider µ1pB0 � a�� bq �µ1pB0q,

µ1pB0 � a�� bq �µ1pB0q � µ1pB�� a�� a0 � b� b0q �µ1pB�� a0 � b0q

¥�µ1pB�� a0 � b0q dµ

1pB�� a�� bq

`�µ1pB�� a0 � bq dµ1pB�� a�� b0q

�µ1pB�q

�µ1pB�� a0 � b0q

¡ 0, (4.3)

where the first inequality follows by applying the exchange condition to the pair pB��a�� a0 � b� b0, B�q and a� R B�. This yields that,

µ1pB�� a�� a0 � b� b0q dµ1pB�q

¥�µ1pB�� a0 � b0q dµ

1pB�� a�� bq�`�

µ1pB�� a0 � bq dµ1pB�� a�� b0q�

The positivity of expression (4.3) is a consequence of the choice of b0 and the choice

of τ.

The proof of the equivalence of (i) and (iii) is elementary. To see that (i) implies

(iii), let X and Y be subsets of R satisfying |X | � d�1 and Y � d�1. Suppose that the

minimum in (4.2) in attained at i�. Then, by the valuated exchange condition stated


in (4.1) applied to X � i�, Y � i� and i�, there exists j� P Y zX with j� � i� such that

µpX � i�q d µpY � i�q ¥ µpX � j�q d µpY � j�q. By choice of i� right and left hand

side of the inequality are equal, showing that the minimum is also attained at j�.

For the converse, (iii) implies (i), let A, B � R such that |A| � |B| � d, and a P AzB.

Consider (4.2) with X � A� a and Y � B � a. Let b P BzA such that µpA� a� bq dµpB� a� bq is minimal among the terms in (4.2). Such b exists by the premise that

the minimum is attained twice. Then µpAq dµpBq ¥ µpA� a� bq dµpB� a� bq.

We obtain the equivalence of (ii) and (iv) by restricting this proof to the collection

of pairs of subsets described parts (ii) and (iv) of the statement.

l

All characterizations above are analogous to various axiomatizations of matroid

in terms of their bases. Items (i)-(iv) differ in the scope and representation of the

exchange axiom. Characterization (vi) is of a different nature. It is perhaps best seen

in light of Gale’s description of matroid basis sets in terms of unique maxima [26]. A

number of different axiomatizations of valuated matroids will appear later in this text,

the first of which is strongly connected to certain submodules associated to valuated

matroids called tropical linear spaces. It appears in the end of the following section.

The characterization (vi) associates combinatorial data to a valuated matroid in

terms of a polyhedral subdivision. This polyhedral subdivision is referred to as the

combinatorial type of the valuated matroid.

Tropical linear spaces

The main reason for studying valuated matroids is that they parameterize nice tropical

prevarieties, called tropical linear spaces. Let µ be a valuated matroid on R over F .

The tropical linear space associated to µ will be defined as a certain finitely generated

subsemimodule of FR. The reason for introducing it this way instead of as a tropical

prevariety is twofold. First, it makes the relation between tropical linear spaces (for

valuated matroids) and lattices of flats (for matroids) clearer from the outset, and

secondly, it allows us to define the prevariety structure using the dual matroid. This

will be made more clear later. It is also more in line with the algebraically flavored

approach to tropical linear spaces that is preferred in this thesis. See also [49].

Let I � R be an independent set of µ of corank 1 and define vµI P FR to be the

point with i-th coordinate µpI � iq, when i R I , or 8, when i P I . When the reference

to the valuated matroid is clear from context we simply write vI . The next lemma is

about minimal sets of generators for the F -module generated by the vectors vI .

Lemma 4.1.4 Let I be a corank 1 independent set of matpµq and let J be a collectionof such sets. The following statements hold:


(i) the point vI is in the tropical linear hull of tvJ | J P J u if and only if the hyperplaneof µ spanned by I equals the hyperplane of µ spanned by J, for some J P J ,

(ii) the support of vI is RzH, where H is the hyperplane spanned by I in matpµq.

Proof. Part (ii) of the statement is immediate: Let I be an independent corank 1 set in

matpµq. Then i P clpIq is equivalent to I� i is not a basis of matpµq. Thus, µpI� iq � 8

if and only if i P clpIq.

For part (i), suppose that there exists J P J such that clpIq � clpJq. By part (i),

vI and vJ have equal support. If the size of the support is at most 1 the implication is

trivially true. Suppose that it is at least two. Let i, j P supppvIq � supppvJq. Consider

(4.2) with Y � I Yti, ju and X � J . If k P I , then k P clpIq and hence k P clpJq. Thus,

the expression reduces to, µpI� iqdµpJ� jq`µpI� jqdµpJ� iq. Since the minimum

is attained twice, µpI � iq dµpJ � jq � µpI � jq dµpJ � iq, which shows that vI and

vJ are proportional.

Suppose that for each J P J there exists λJ P F such that,

vI �àJPJλJ d vJ .

Let J be such that clpJq � clpIq. Since any two hyperplanes are incomparable in �-

order, there must exist i P clpIqz clpJq. Thus, J� i is a basis, and I� i isn’t and λJ �8.

Thus, vI is already in the module generated by all vJ with clpJq � clpIq and J P J . In

particular, there exists J P J such that clpIq � clpJq. l

By the above lemma, any minimal set of generators consists of a set of vectors

tvI | I P J u subject to the condition that every hyperplane of µ is the closure of

exactly one I P J . To avoid a choice for a specific set of generators we associate to

every hyperplane H the point vH defined as the sum of the vI , where I runs over all

corank 1 subsets that span H. By the lemma, it holds that vH � vI for some I . This

allows for a clean definition of the tropical linear space as a tropical linear hull.

Definition 4.1.3 Let µ be a valuated matroid on R over F . The tropical linear space of

µ is the underlying set of the F -module generated by tvH | H is a hyperplane of µu. It

is denoted Lµ.

Example. Let n P N and µ a valuated matroid of rank 2 over R8, such that µpBq 8

for each B P P2rns. Then the underlying matroid of µ is the uniform matroid of rank

2 on n. The hyperplanes of the latter matroid are exactly the singletons tiu and the

corresponding generator is

vtiu � pµpti, 1uq, . . . ,µti� 1, iu,8,µti, i� 1u, . . . ,µti, nuq.


The tropical linear spaceLµ has topological dimension two (we will prove this later),

and hence its projectivization has topological dimension 1. This projectivization is a

tree on n leaves, with the i-th leave being the projectivization of vtiu.

The next proposition makes clear that the tropical linear space associated to the

valuated matroid is an analogue of the lattice of flats of the matroid. It shows in fact

that if F � B, or equivalently, when µ is an ordinary matroid the tropical linear space

is isomorphic the lattice of flats of µ.

Proposition 4.1.5 (Immediate properties) (i) The tropical linear space Lµ has thestructure of a lattice under the partial order restricted from FR. (ii) The function thatassigns Rz supppvq to each point v P FR is a lattice morphism that maps Lµ surjectivelyonto the lattice of flats of the underlying matroid of µ.

Proof. Part (i) is an immediate consequence of the fact that a finitely generated sub-

module of FR is a lattice (cf. Proposition 3.1.6). Part (ii) follows from Lemma 4.1.4

and the fact that supppv`wq � supppvq X supppwq. l

The following is a characterization of valuated matroids in terms of tropical de-

pendence (cf. Section 3.3).

Lemma 4.1.6 Let µ : PdpRq Ñ F be a function such that the set tA P PdpRq | µpAq �8u forms the set of bases of a matroid M. Then µ is a valuated matroid if and only if foreach triple I , J , K of independent (in M) sets of size d�1 in such that I X J XK has sized � 2 the set tvI , vJ , vKu is tropically dependent.

Proof. Suppose µ is a valuated matroid and set S � IXJXK . Then |S| � d�2, where

d � rkµ. Write IzS � tiu, JzS � t ju and KzS � tku. We need to show that there

exists x I , xJ , xK P F such that

px I dµpSilqq ` pxJ dµpS jlqq ` pxK dµpSklqq, (4.4)

vanishes tropically for every l R S. Set x I � µpS jkq, xJ � µpSikq and xK � µpSi jq.The expression vanishes by characterization (v) of Theorem 4.1.3.

Conversely, let S � R with |S| � d � 2 and i, j, k, l P RzS distinct. Set I � Si,J � S j and K � Sk. By the condition on the tropical rank there exist x I , xJ , xK P Fsuch that the expression px I d vIq ` pxJ d vJq ` pxK d vKq vanishes tropically, or in

other words, that the minimum of (4.4) is attained at least twice. Without loss of

generality, suppose that x I � µpS jkq. Then, by the expression for l � k it follows that

xJ � µpSikq. Setting l � j gives xK � µpSi jq. Hence, µ satisfies (v) of Theorem 4.1.3.

l


Examples. We end this section with a number of typical examples of tropical linear

spaces.

(i) Let R be a finite set. Then FR is a tropical linear space. The proportional equiva-

lence class of valuated matroids on R is that of the free valuated matroids on R,

i.e. those valuated matroids on R of rank |R|. The hyperplanes of this matroid

are all sets of the form Rztiu, and vRztiu equals ei , i.e. the vector all of whose

entries are 8, except the i-th, which is 0.

All tropical linear spaces are by definition contained in some free F -module.

Since they are also invariant under tropical scalar multiplication no information

is lost by drawing the projectivizations, instead of the actual tropical linear

spaces. The picture one should have in mind of FR is the |R| � 1 dimensional

simplex (at least if F � R8, a subset thereof if F is strictly contained in R8).

The reason is that PpRR8q is homeomorphic to the |R| � 1-dimensional simplex.

Distances, however, are of course distorted. To clarify matters we have drawn

R48 and B3 in Figures 4.3(a) and 4.3(b).

(∞,∞, 0,∞)

(∞, 0,∞,∞)

(0,∞,∞,∞)

(∞,∞,∞.0

(a) (Projectivization of) four dimensionaltropical affine space over R

8.

(0, 0, 0)

(0, 0,∞)

(0,∞, 0)(∞, 0, 0)

(∞,∞, 0)

(0,∞,∞)(∞, 0,∞)

(b) (Projectivization of) three dimen-sional tropical affine space over B.

Figure 4.3: Tropical affine spaces.

(ii) Any geometric lattice embedded as a meet-subsemilattice in the full power set

lattice of a finite set is a tropical linear space. The full power set lattice is the

ambient tropical affine space. Equivalently, the underlying set of the Bergman

fan of a matroid is the intersection of a tropical linear space in Rn8 with Rn [2].

(iii) More specific examples of (projectivizations of) tropical linear spaces are given

after the prevariety structure on a tropical linear space is dealt with, in Section

4.1.2

4.1.2 Basic constructions and relations

This section deals with various ways of constructing new valuated matroids from a

set of given ones: orthogonal dual, restriction, contraction, direct sum, matroid union


and matroid intersection. A number of these constructions have definitions which are

more easily stated in terms of an equivalent, or cryptomorphic characterization of val-

uated matroids in terms of a function on independent sets of the underlying matroid.

For valuated matroid this characterization is stated and proved to be cryptomorphic

by Murota in [48]. We follow his exposition.

Let R be a finite set and ν : P pRq Ñ F a function. Then ν is said to be a valuatedmatroid on independent sets if the following conditions are satisfied.

(i) ν is not identically 8,

(ii) ν is increasing, i.e. νpIq ¤ νpJq if I � J ,

(iii) if I is a proper subset of J and νpJq � 8 then there exists i P JzI such that

νpJq � νpI � vq,(iv) if |I | � |J | � 1, there exists v P JzI such that νpIqd νpJq ¥ νpI � vqd νpJ � vq.

The collection tI � R | νpIq � 8u form the collection of independent sets of an (or-

dinary) matroid on R and hence the maximal sets of this collection are equicardinal,

of, say, cardinality d. The translation from valuated matroids on independent sets

to the valuated matroids of Definition 4.1.1 is by restricting to PdpRq. The inverse

translation is µ ÞÑ µ, where,

µpIq �à

BPBpµqI�B

µpBq,

and Bpµq is the set of basis of matpµq. Definition 4.1.1 and the independent set

characterization are cryptomorphic in the sense that the restricting to PdpRq subsets

with finite image and µ ÞÑ µ are inverse bijections between the valuated matroids on

independent sets and the valuated matroids of Definition 4.1.1.

Truncation

The first construction of this section is strongly related to the independent set axiom.

One can wonder whether a valuated matroid on independent sets, restricted to sets

of a certain cardinality is a valuated matroid, if it is not identically 8. The following

lemma, combined with an easy induction show that this is indeed the case.

Lemma 4.1.7 Let µ : PdpRq Ñ F be a valuated matroid. The map Pd�1pRq Ñ Fdefined by,

A ÞÑàA�B

BPBpµq

µpBq, A PPd�1pRq,

is a valuated matroid.

Proof. For a proof the reader is referred to [48]. l

The matroid in the lemma is referred to as the truncation of µ and denoted Tpµq.It is the valuated analogue of the truncation of an ordinary matroids (cf.[11, Section


7.4]).

Orthogonal dual

The characterization of the tropical linear space as a tropical prevariety, i.e. an inter-

section of tropical hyperplanes is conveniently stated in terms of the orthogonal dual

of the valuated matroid. The definition is straightforward.

Definition 4.1.4 The orthogonal dual of µ is the function µK : P pRq Ñ F defined

by µKpX q � µpRzX q. When confusion is unlikely to occur, we will omit the adjective

orthogonal and refer to it simply as the dual of µ.

Lemma 4.1.8 The orthogonal dual is a valuated matroid of rank |R| � rkµ.

Proof. Let A, B � R satisfy |A| � |B| � |R| � rkµ and a P AzB. Then a P pRzBqzpRzAqand hence there exists b P pRzAqzpRzBq such that,

µKpAq dµKpBq � µpRzAq dµpRzBq

¥ µpRzA� a� bq dµpRzB� a� bq

� µKpA� a� bq dµKpB� a� bq.

l

Clearly, pµKqK � µ. For matroids over B the dual is precisely the orthogonal dual

of an ordinary matroid. The relation between the tropical linear space of a matroid

and its dual is given in the next proposition. To state it we define the map x�, �y :

FR � FR Ñ F by,

xv, wy �àiPRpvi dwiq, v, w,P FR.

This is a tropical bilinear form in the sense that both v ÞÑ xv, by and w ÞÑ xa, wy are

linear functions on FR. The set of all w P FR for which the expression xv, wy tropically

vanishes (i.e. it equals 8, or the minimum of the terms is attained at least twice) is

denoted vK. Similarly, for any set V � FR we set,

VK � tw P FR |À

iPR vi dwi vanishes tropically for all v P Vu

Proposition 4.1.9 Let µ be a valuated matroid on R. Then LKµ �LµK .

Proof. We start by proving the inclusion LµK �LKµ . The rank of µK is |R| � d, where

d is the rank of µ. It is sufficient to prove that for any set I of size |R| � d � 1 the


vector αI � pµKpI � iqqi is tropically perpendicular to Lµ. Let J � R be a set of size

d � 1. Consider pαI , vJq,

pαI , vJq �àiPRµKpI � iq dµpJ � iq

�àiPRµpRzI � iq dµpJ � iq

�àiPRzI

µpRzI � iq dµpJ � iq.

The minimum of this expression is attained twice by (4.2).

To prove the converse the following lemma is required. For v P FR define µJ to be

the function P psupp vq Ñ F ,

Z ÞÑ vZXsupp v dµpZ Y Jq,

where vX denote the tropical product of the coordinates of v indexed by X and J is

an independent set of matpµq such that supp v Y J is spanning in matpµq. This is a

valuated matroid again, and hence the collection of arguments on which it is minimal

(its residual matroid) is the collection of bases of a matroid. We prove that if v KLµthen respµJq has no isthmuses, where an isthmus is a point i P R that is in every basis

of µ. Suppose that it has an isthmus j P R. Let A P respµJq be a basis. Consider the

expression,

vIXsupp v dàiRIpvi dµppA� jq Y J � iqq

By definition, the minimum of this expression is attained at j. Suppose that it were

also attained at i. Then vi �8. But i R supp v, since j is an isthmus of µJ .

Let v PLKµ . We need to show that,

v �à

Spv : wSq dwS ,

where |S| � d � 1 and wS has j-th coordinate µpS � jq. These are the canonical

generators of LµK . Let j P R and suppose that v j � 8. In case v j � 8 the j-thcoordinate of the expression on the right hand side is also8, and we are done. Choose

A and J such that respµJpAqq is minimal among all such choices, with j R A. This is

possible, since no µJ has isthmuses. Set S� � AY J � j. We claim that,

v j � pv : wS�q dwS� � pv : wS�q dµpS�� jq.


This translates to,

v j � suptvi �µpS�� iq | i P S�,µpS�� iq � 8udµpS�� jq.

It is thus sufficient to prove that the maximum over all vi �µpS�� iq, with i P S� and

µpS� � iq � 8, is attained at j. This is a consequence of the construction of S�. This

proves the statement. l

The proof is a bit technical, but this is due to the limited mathematical machinery

we have at our disposal at this point. Later, in Section 4.2.1 more geometric insight is

given.

Corollary 4.1.10 The tropical linear space Lµ is the intersection of the hyperplanesassociated to αRzC , where C runs over the circuits of µ.

Proof. By the previous proposition, Lµ �LKµK

. Thus, Lµ is the collection of points in

FR that are tropically perpendicular to the hyperplane vectors αH of LµK , where Hruns over the hyperplanes of µK. The statement follows by the fact that the setwise

complements of the hyperplanes of the dual are exactly the circuits of µ. l

The above corollary is the usual definition of a tropical linear space, as found in

[59]. We give some more specific examples of tropical linear spaces.

Example.

(i) Tropical affine space F n is the tropical linear space associated to the matroid µ

on R that assigns 0 to R. Its dual is the matroid that assigns 0 to H, which has

no hyperplanes. Hence F n is the tropical prevariety of H.

(ii) A more specific example is the tropical linear space L associated to µ on r4s

over R8, where µ has underlying matroid the uniform matroid of rank 3 on

r4s. It consists of 6 cones that intersect in the principal submodule generated

by p�µp234q,�µp134q,�µp124q,�µp123qq, where � denotes the the ordinary

minus on R. Its projectivization is depicted in Figure 4.4(a). The white dots on

the boundary of the simplex represent the canonical generators. Equivalently,

it is the subset of R48 perpendicular to LµK , which is the module generated by

vµK

H � pµp234q,µp134q,µp124q,µp123qq.

(iii) A function µ :P2r4s Ñ R8 is a valuated matroid when the expression,

pµp12q dµp34qq ` pµp13q dµp24qq ` pµp14q dµp23qq, (4.5)

vanishes tropically. If matpµq � U2r4s, the hyperplanes of µ are the singletons,

and Lµ � xvt1u, vt2u, vt3u, vt4uy. Equivalently, L is the locus of points perpen-


(a) Projectivization of a tropicallinear space associated to a valu-ated matroid with underlying ma-troid U3,4.

v{2}

v{1}

v{4}

v{3}

(b) Projectivization of the tropi-cal linear space associated to avaluated matroid µ with underly-ing matroid U2,4.

Figure 4.4: Projectivizations of tropical linear subspaces of R48.

dicular to LµK . The latter is generated by the elements,

wt1u � p8,µp34q,µp24q,µp23qq

wt2u � pµp34q,8,µp14q,µp13qq

wt3u � pµp24q,µp14q,8,µp12qq

wt4u � pµp23q,µp13q,µp12q,8q,

and hence L is the intersection of the four hyperplanes wKtiu. It is depicted in

Figure 4.4(b) in case µp14q dµp23q is the unique maximum in (4.5).

Restriction and contraction

The operations of restriction, and its dual, contraction, have analogues for valuated

matroids. They were already given in [23] for equivalence classes of valuated ma-

troids, but with minor modifications their constructions lift to the individual valuated

matroids themselves. In this subsection µ is a valuated matroid on R over a tropical

semifield F .

Let X � R. The restriction of µ to X is the function on subsets of X that assigns to

each J � X the minimum of µpZq, where J � Z � R. It is denoted µrX s. Contractionis the dual of this operation. i.e., the contraction of µ to X is pµKrX sqK. It is denoted

µ � X , or, when more convenient, by µ{pRzX q. The fact that both operations indeed

yield valuated matroids and the characterization of their tropical linear spaces is given

by the next lemma.


Lemma 4.1.11 Let µ be a valuated matroid on R and X � R. Then both µrX s and µ � Xare valuated matroids on X . Moreover,

(i) the tropical linear space associated to the restriction of µ to X is the image of Lµunder the natural projection FR Ñ F X ,

(ii) the tropical linear space associated to the contraction to X is the image of thetropical linear space tv P Lµ | vi �8 for all i R Xu under the natural projectionFR Ñ F X .

Proof. We prove the statements in the numbered order.

(i) We should start by showing that the restriction is indeed a valuated matroid.

Choose I a set of size rkµ R � rkµ X such that I Y X is spanning. Denote the map

P X Ñ F defined by A ÞÑ µpAY Iq by µI for the purpose of this proof. That µI is a

valuated matroid is a direct consequence of the exchange condition. Moreover, µrX sis the pointwise tropical sum (minimum) of the µI . To show that µrX s is a valuated

matroid it is sufficient to show that the µI for varying I are proportional. For this,

assume without loss of generality that rkµ R� rkµ X � 1. Let i, j P RzX such that both

X � i and X � j are spanning sets of µ and let A, B � X be sets of size rkµ R� 1. We

need to show that µpA� iq dµpB� jq � µpA� jq dµpB� iq.Suppose that one of the terms is 8. Assume without loss of generality that µpA�

iq � 8. Then A� i is not a basis of matpµq and hence A does not span X . Thus, A� jis not a basis of matpµq, or, equivalently, µpA� jq � 8.

Suppose that none of the terms is is 8. Consider (4.2) with Y � AY ti, ju and

X � B. If k P A, then B � k � X and hence µpB � kq � 8. The required equality

µpA� iqdµpB� jq � µpA� jqdµpB� iq follows since (4.2) now has only two terms,

and the minimum must be attained at least twice. Thus, we have shown that the µI

are proportional.

We now show that the projection contains the tropical linear space associated to

µrX s. Let H be a matroidal hyperplane of µrX s. Then H � X X G for some matroidal

hyperplane G of µ (cf. [11, Proposition 7.3.1. on page 131]). There exists a spanning

set I for G such that IXH spans H. Thus, the rank of IzH is |I |�rkµ H � rkµ R�rkµ X .

Set J � IzH. Then, πpvµI q � vµJJ , where the superscript denote the matroid with

respect to which the generating vector is defined. Thus, πpvµHq is proportional to vµrX sG ,

proving the inclusion.

To prove the other inclusion, we make use of Corollary 4.1.10. Let v P πX pLµq and

C a circuit of µrX s. Then C is also a circuit of µ (cf. [11, Proposition 7.3.1. on page

131]). Let S � C be a minimal dependent spanning set of µrX s such that αC � αS .

Let I � R such that µrX spAq � µpAY Iq. Then S Y I is a minimal dependent spanning

set of µ containing C , and hence the minimum,

àiPCµpSY I � iq d vi �

àiPCµrX spS� iq d vi


is attained twice, proving that v PLµrX s.(ii) That the contraction is a valuated matroid is a direct consequence of part (i)

of this Lemma and Lemma 4.1.8. For the purpose of this proof, write L8 for the set

tv | vi �8 for all i R Xu and denote the canonical coordinate projection FR Ñ F X by

π. We prove that πpL8q �Lµ�X .

Note that Lµ�X � πpLµKqK by Proposition 4.1.9 and part (i) of this lemma. Thus,

it is sufficient to show the statement that v K w is equivalent to πpvq K πpwq, for

any v P L8 and any w P FR. Let v K w. Then the minimum of xv, wy is attained at

least twice. If xv, wy � 8, then the minimum xπpvq,πpwqy is attained at least twice;

if xv, wy � 8, then xπpvq,πpwqy � 8. Proof of the converse is similar. l

This section ends with a useful corollary of the proof.

Corollary 4.1.12 There exists I�, J� � R such that I�Y X is spanning in µ and J� is abasis of µzX such that µrX spZq � µpZ Y I�q and µ{X pZq � µpZ Y J�q, for all Z � R ofthe appropriate size.

Direct sum

The next construction is very simple and natural from the point of view of tropical

linear spaces.

Proposition 4.1.13 Let µ and ν be valuated matroids on disjoint sets R and S over F.Then the function,

P pR\ Sq Ñ F, X ÞÑ µpX X Rq d νpX X Sq

defines a valuated matroid on R\S over F. Its associated tropical linear space isLµ�Lν .

Proof. The valuated exchange condition is easily seen to follow from either the valu-

ated exchange condition for µ or that of ν .

The corank 1 independent sets of µ` ν are of the form A\ J or B \ I , where Iis a corank 1 independent set of µ, J is a corank 1 independent set of ν , A is a basis

of µ and B is a basis of ν . The fundamental generators associated to these sets are

p8, . . . ,8q�µpAq d vνJ and νpBq d vµI � p8, . . . ,8q. They also generate Lµ�Lν . l

The valuated matroid defined in the above proposition is called the direct sum of µ

and ν and denoted µ`ν . It generalizes the direct sum operation on matroids (cf. [11,

Section 7.6.]) in the sense that matpµ`νq �matpµq`matpνq, where the `-operator

on the left hand side is the direct sum as defined above, and the `-operator on the

right hand side is the direct sum of ordinary matroids.


Union and intersection

A construction that is important in the proof of Proposition 4.2.21 is that of valuated

matroid union. This construction is dual to the matroid intersection construction on

valuated matroids that appears in [59] and is the valuated equivalent of an earlier

matroid construction [11]. The construction assigns to valuated matroids µ and ν on

a common ground set R another valuated matroid, µY ν on R by setting,

µY νpZq �à

XYY�ZµpX q d νpY q,

in the independent set characterization. When at least one basis µ is disjoint from a

least one basis of ν , then the definition in terms of the bases cryptomorphism simpli-

fies to the following,

pµY νqpZq �à

X\Y�ZµpX q d νpY q.

That both these definitions in fact define the same valuated matroid is stated and

proved in the next proposition.

Proposition 4.1.14 The union µ Y ν is a valuated matroid on R that is dual to thevaluated matroid intersection of µ and ν .

Proof. We simplify the situation somewhat to be able to make use of the results in

[59]. First of all, note that it is sufficient to prove that µ Y ν defines a valuated

matroid in case that some basis of µ is disjoint from some basis of ν . Since, suppose

that this is not the case, then there exist truncations µ1 of µ and ν 1 of ν such that

µY ν � µ1Y ν 1 and µ1 and ν 1 have disjoint bases.

We proceed by proving that µY ν equals pµKX νKqK. Let Z P R. Then,

pµKX νKqKpZq �à

XXY�RzZ

µKpX q d νKpY q

�à

XXY�RzZ

µpRzX q d νpRzY q

�à

RzXYRzY�Z

µpRzX q d νpRzY q

�à

X 1YY 1�Z

µpX 1q dµpY 1q.

The statement follows by Proposition 3.1. of [59]. l

Note that direct sum is a special case of the union. Let µ and ν be valuated ma-

troids on R and S over F , with R and S disjoint sets. We write υ0pX q for the valuated

matroid on X of rank 0 that assigns 0 to H. Then µ` ν � pµ`υ0pSqqY pν `υ0pRqq.


Extensions and quotients

The inclusion relation on tropical linear spaces contained in the same ambient space

is reflected in the quotient relation between valuated matroids on the same ground

set. Let R be a finite set. Let µ and ν valuated matroids on R over F . Then ν is said

to be quotient of µ when Lν � Lµ. We write ν � µ when ν is a quotient of µ. By

Proposition 4.1.9 it is immediate that ν � µ if and only if µK � νK.

Concluding remarks

In sections 4.2 and 4.3 of this chapter the focus is on the tropical linear space as-

sociated to a valuated matroid, rather than on the valuated matroid itself. To avoid

having to define the appropriate maps every time we speak of the tropical linear space

associated to a construction on a valuated matroid, we apply the notations of certain

constructions to the tropical linear space as well. For example, instead of writing the

projection of Lµ to FS , where µ is a valuated matroid on a finite set R and S � R, we

write L rSs. Where there already exists good notation for the linear spaces associated

to constructions (e.g. in the case of the direct sum), we always prefer the old notation.

The convention is mostly useful for restrictions and contractions, and even in these

case, we will not always use it. Note that by Proposition 4.1.9 this notational con-

vention agrees with the orthogonal dual of the tropical linear space, that we defined

earlier.

4.1.3 Parameter spaces

Let L be a tropical linear space in FR. A tropical linear spaceM in FR is said to be

a subspace of L ifM � L . The collection of tropical linear subspaces of dimension

d of L is denoted DrdpL q, and the collection of all linear subspaces of L is DrpL q.Following [33], we refer to these sets as Dressians, after Andreas Dress. By definition

of a tropical linear space, the collection DrdpL q can be embedded as a subset of the

collection of proportional equivalence classes of valuated matroids on R over F . This

is the content of Proposition 4.1.16, which we prove after the next lemma.

Lemma 4.1.15 Let µ and ν be valuated matroids on R over F. Then Lµ � Lν if andonly if µ and ν are equivalent.

Proof. If µ and ν are equivalent, there exists λ P F� such that µ� λdν . In particular,

they have the same underlying matroid and rank, say d. Let I be a set of cardinality

d � 1. Then vµI � λd vνI � Lν and vice versa. The statement follows since the vµIgenerate Lµ as an F -module.

For the other implication, suppose that Lµ �Lν . By part (ii) of Proposition 4.1.5

the lattices of flats of matpµq and matpνq are the same and hence matpµq and matpνq


are the same matroid M . By considering the supports one sees that vµH and vνH are

proportional for all hyperplanes H of M . By Proposition 4.1.4 for all I of cardinality

d�1 there exists λI P F such that vµI � λI d vνI . We show that µpAq�νpAq is constant

for all bases A of M , by induction on the symmetric distance between bases.

Let A and B be bases of M such that |A ∆ B| � 2. Write I � AX B, A� I � i and

B � I � j. Since vµI � λI d vνI it follows that µpAq � λI d νpAq and µpBq � λI d νpBq.Let A and B be bases of M and suppose that for all bases C and D such that

|A∆ B| ¡ |C ∆ D| it holds that µpCq�νpCq and µpDq�νpDq are equal, to say λ P F�.

Since, by the basis exchange axiom for ordinary matroids, there exists a basis C� such

that |A∆ C�|, |B ∆ C�| |A∆ B| the statement follows. l

This lemma has the interesting consequence that the space of all linear subspaces

of a fixed dimension d of some ambient affine space is essentially a projective tropical

prevariety in the projectivization of the space of maps PdpRq Ñ F . As a matter of

fact, in this thesis the Dressian shall be identified with this embedding whenever it is

convenient to do so.

Proposition 4.1.16 Let L � FR be a tropical linear space. The Dressian DrdpL q em-beds as a tropical prevariety in DrdpF

Rq. In particular, it embeds as a tropical prevarietyin the projectivization of the space of maps PdpRq Ñ F.

Proof. Let µ be a valuated matroid associated to L . The Dressian DrdpL q is the

space of equivalence classes of valuated matroids µ on R over F of rank d satisfying

Lν � L . Thus, it is clearly a subset of DrpFRq. We need to show that the condition

Lν � L is given by a the vanishing of a finite number of tropical polynomials. By

Proposition 4.1.9 the condition is equivalent to Lν K LK � LµK . By linearity this

condition is equivalent to the perpendicularity of a set of generators of Lν and a set

of generators of LµK , both of which can be taken finite. Thus, the statement follows.

l

A question related to parameter spaces which deserves mention is the following.

Let K denote a field with surjective valuation v : K Ñ F and consider a linear space

L � KR with tropicalization L . Tropicalization maps GrdpLq to DrdpL q, but is in

general not surjective. In fact, the image depends on the specific choice of field Kand L such that vpLq � L . Determining the image is a difficult question, unless

d � 2 and K is algebraically closed. The general question reduces to determining

when a matroid is realizable over a K , in case v is the trivial valuation and F � B. To

show that in general the Dressian can be expected to have a much bigger dimension

that the image of the Grassmannian under the tropicalization map, we state the next

proposition, first stated and proved in [33] (as Theorem 3.6).

Proposition 4.1.17 (Herrman, Jensen, Joswig, Sturmfels) Let K be an alge-


braically closed field with surjective valuation ω : K Ñ R8. The dimension of Dr3pRn8q

is of order Θpn2q, while Gr3pKnq, and hence ωpGr3pK

nqq, has dimension 2n� 9.

Remark. It is worthwhile to mention here the well-known fact that ωpGr2pKnqq �

Dr2pRn8q, i.e. that all projectivized tropical linear are realizable. However, we later

give the explicit example Dr2pL q, with L the tropical linear space of µ of rank 3

on r4s that assigns 0 to each of its bases. To do so, it is necessary to study functions

on tropical linear spaces. An interesting feature of the example is that this particular

Dressian is actually a tropical linear space of dimension 3 and hence, that not all lines

in L are realizable as a line in a fixed realization of L . Or in other words, Dr2pL q is

not the tropicalization of (an embedding of) the Grassmannian Gr2pK3q.

4.2 Functions and their graphs

This section is devoted to studying functions from a tropical linear space to the semi-

field the space is defined over. The collection of all such functions that are linear is the

algebraic dual of the tropical linear space, as defined in Section 3.2. We use Lemma

3.2.1 to derive that linear functions are the restrictions of homomogeneous polyno-

mial functions of degree 1. This is not specific to tropical linear spaces, but holds

for any submodule of FR. However, the tropical linear space, through its valuated

matroid, defines a polyhedral structure on the homogeneous polynomial functions of

degree 1, which is intimately related to restrictions of these functions to the tropical

linear space. The main statement is Proposition 4.2.2. It has as a consequence that dterms are sufficient to specify the restriction to a tropical linear space of rank d. In

Theorem 4.2.3 a subcomplex of the polyhedral structure is shown to have underlying

set equal to the dual tropical linear space. This statement is essentially the character-

ization of a tropical linear space stated in Proposition 2.3 of [59].In the second part of this chapter we classify the elementary extension functions

on a fixed tropical linear space, i.e. the functions whose graph, defined in a suitable

manner (we refer to it as the extended graph), is again a tropical linear space. The

theory developed in the first of part of Section 4.2.2 is an extension of that devel-

oped by H. Crapo on elementary extensions of matroids in [14]. Our main results are

that, although such functions need not be linear, they are defined by their image on

the canonical generators of the tropical linear space. This is the content of Theorem

4.2.11. Moreover, the collection of all such functions forms a module, that is also a

tropical prevariety in a natural way. The exact result is stated in Theorem 4.2.9 and

Corollary 4.2.10. These are the main result. The remainder of the second part of this

chapter is devoted to examining elementary extension functions in more detail. In

particular, we prove that the linear functions are always elementary extension func-

4.2. FUNCTIONS AND THEIR GRAPHS 83

tions and that they are the best-behaved, both with respect to realizability over a field

(Proposition 4.2.15), and with respect to linear subspaces (Proposition 4.2.17).

The third part is the second section of Section 4.2.2, and aims to understand the

situation in which we iteratively take the extended graph of a linear function. More

precisely, if L over F is a tropical linear space, and α : L Ñ F is a linear function,

then the extended graph L1 is a tropical linear space. The extended graph of a linear

function α1 :L1 Ñ F is again a tropical linear space L2. Repeating the process yields

a sequence of tropical linear space that we represent as,

Lα1 // L1

α2 // L2α3 // . . . αk // Lk (4.6)

This situation can be usefully represented by a weighted directed graph on R Ytp1, . . . , pku nodes, with R such that L � FR. The valuated matroid of Lk is de-

termined by that of L and the weighted directed graph, or rather by its valuatedlinking system, i.e. the weights of a certain collection of vertex disjoint paths. A gen-

eral valuated linking system is an abstraction of this concept, and they are used as a

tool in the second section of Section 4.2.2 to understand the situation represented by

(4.6). The linking systems that come from weighted directed networks are then used

to construct the morphisms in the category of tropical linear spaces.

4.2.1 Linear functions

This section deals with the algebraic dual of a tropical linear space, as defined in

Section 3.2. We remind the reader that the map that assigns the module of linear

functionals to a tropical linear space, and the pullback to a homomorphism between

tropical linear spaces, is a contravariant functor on the subcategory of tropical linear

spaces and homomorphisms. The following statement is essential for our analysis,

and is an immediate consequence of Lemma 3.2.1.

Lemma 4.2.1 Let L � FR be a tropical linear space. Any linear functional in L � is therestriction of a linear functional FR Ñ F. Or more precise, for each φ P L �. there existpλiq P FR such that for all v PL ,

φpvq �¸iPR

λi d vi . (4.7)

Proof. By Lemma 3.2.1 there exists a linear functional ψ : FR Ñ F such that ι�pψq �

φ, where ι :L ãÑ FR is the canonical injection. In other words, φpvq �ψpvq for each

v PL . Denote the i-th canonical free generator of FR by ei . Thenψpvq �°

iPRψpeiqd

vi , which proves the second statement. l


(∞, 0,∞)

(∞,∞, 0)

CM

(0,∞,∞)

{{2, 3}}

{{1, 2}}

{{1, 3}}

Figure 4.5: The polyhedral subdivision of Q38ÑQ8 induced by the matroid U2pr3sq.

The matroids associated to the maximal cones are specified by their collection ofbases.

Let µ be a valuated matroid on R over F with associated tropical linear space

Lµ � FR. For each linear functional φ : FR Ñ F define the support of φ as the set of

i P R for which φpeiq � 8 and denote it by suppφ. In the case that suppφ � R the

vector�φpeiq

�lies in FR

� and hence the twisted matroid�φpeiq

��µ, with

��φpeiq

��µqpBq �

�àiPBφpeiq

dµpBq (4.8)

is well-defined. We denote it simply by φ �µ.

Write S for the support of φ and denote the injection FS Ñ FR that maps the

i-th canonical generator of FS to the i-th canonical of generator of FR by ι. Then the

support of ι�pφq is S and hence ι�pφq �µrSs is well-defined. Now define the ordinary

matroid µφ as the direct sum of matpµ{Sq and respι�pφq �µrSsq, i.e.,

µφ �matpµ{ suppφq ` respι�pφq �µrSsq. (4.9)

The map φ ÞÑ µφ induces a subdivision of the space of linear functionals FR Ñ F . We

give an example below.

Example. Consider the valuated matroid µ on r3s over Q8 with µpBq � 0 for all

B P P2pr3sq. The association φ Ñ µφ defines a polyhedral subdivision of the space

of linear functionals Q38 Ñ Q8. We identify this space of linear functionals with F3

through the natural map φ ÞÑ pφpe1q,φpe2q,φpe3qq. The polyhedral subdivision of

PpQ8q induced by the above association is drawn in Figure 4.5.

As a first example, consider the linear map φpz1, z2, z3q � z3, identified with the

point p8,8, 0q PQ38. Its has support t3u and the associated matroid is the direct sum


of matpµq{t3u, which has bases tt1u, t2uu, and resp0 �µrt3usq, which has bases tt3uu.

Thus, M � µφ has collection of bases tt1, 3u, t2, 3uu. We set out to determine which

other functions induce this matroid as well. Note that M is a direct product in two

ways,

M � tt3uu` tt1u, t2uu, M �H`M ,

where we have specified the matroids by their collection of bases. The first decom-

position comes either from points p8,8,φ1q (all of which indeed induce M), or

from points pφ1,φ2,8q, where φ1, φ2 and φ3 are finite. The latter induce a matroid

µ{t1, 2u ` resppφ1,φ2q � µq � H` tt1, 2uu. Remains, the case resppφ1,φ2,φ3q � µqq.

This equals M exactly when,

φ1 �φ3 � φ2 �φ3 φ1 �φ2

in non-tropical notation. This is the cone CM � tpv, v, wq P Q | w vu. It is invari-

ant under classical addition of p1, 1,1q, or, stated differently, under tropical scalar

multiplication and hence has a well defined image in PpQ38q.

There is a strong relation between the matroid µφ and the restriction of φ to Lµ,

which is stated in the next proposition.

Proposition 4.2.2 Let µ be a valuated matroid on R over F and φ : FR Ñ F a linearfunctional. Write S for the support of φ and define the linear map ι : FS

ãÑ FR byιpeiq � ei . Then, for a subset X � R, we have,

φpvq �àiPXφpeiq d vi , for all v PLµ, (4.10)

if and only if X X S is a spanning set of respι�pφq � µrSsq. Moreover, if for each v PLµ � p8, . . . ,8q there is i P X such that φpvq � φpeiq d vi with vi � 8 then X is aspanning set of µφ .

Proof. Since (4.10) is equivalent to φpιpvqq � `tφpeiqdvi | i P XXSu for all v PLµrSsit is sufficient to prove the first statement in the case that S � R, which we assume in

the remainder of its proof.

Let X be a spanning set of respφ �µq, containing a basis B. It is sufficient to prove

(4.10) for the fundamental generators vI , with I an independent set of µ of corank 1.

Consider,

φpvIq �àiPRφpeiq dµpI � iq

and let i� be an index at which the minimum is attained. Suppose that i� R B, since

otherwise the statement is trivial. Then φpeiqI dφpei�qdµpI� i�q is minimal among

the terms of φpeiqI d φpvIq, where we have written φpeiqI to denote the tropical


product dtφpeiq | i P Iu. Note that φpeiqI dφpei�qdµpI � i�q � pφ �µqpI � i�q. The

exchange condition for φ �µ states that there exists j� P BzI such that

pφ �µqpBq d pφ �µqpI � i�q ¥ pφ �µqpB� i�� j�q d pφ �µqpI � j�q.

Since pφ �µqpBq ¤ pφ �µqpB� i�� j�q, by choice of B, the inequality pφ �µqpI� i�q ¥pφ �µqpI � j�q follows. Thus, the minimum in φpvIq is also attained at j� P B.

To prove the reverse implication, let X � R such that φpvq � `tφpeiqd vi | i P Xufor all v PL and let B� be a basis of respφ �µq for which |B�zX | is minimal. Assume,

for the sake of contradiction, that the difference is not empty and let j P B�zX . Clearly,

the minimum of φpvB�� jq is attained at index j. It is also attained at some i P X , by

choice of X . Then i R B� and B� � j � i is a basis of respφ � µq. Moreover, |X zB�| ¡|X zB� � j � i|, contradicting the assumption that B�zX �H. Thus B� � X , showing

that X is a spanning set of respφ �µq.

For the second statement we prove a characterization of the spanning sets of

matpµ{Sq in terms of φ. The claim is that a set X � RzS is spanning in matpµ{Sqif and only for each v P L satisfying vi � 8 for all i P S there exists j P X such that

v j � 8. It is sufficient to prove this statement for v a fundamental generator. Such

fundamental generators correspond to hyperplanes that contain S and the statement

translates to the following: X � RzS is a spanning set of matpµ{Sq if and only if X � Hfor each hyperplane H � S. Since the hyperplanes of matpµ{Sq are of the form HzS,

where H is a hyperplane of matpµq containing S, it is sufficient to show that the span-

ning sets of a matroid are exactly those sets not contained in any hyperplane. But

this equivalence is clear by the fact that both matroid closure and rank are increasing

functions.

A set X � R is spanning in µφ if and only if X X S is spanning in respι�pφq �µrSsqand X zS is spanning in matpµ{Sq. Let v P Lµzp8, . . . ,8q. If supp v X S � H, then

φpvq � `tφpeiqd vi | i P X XSu �8, since X XS is spanning in respι�pφq �µrSsq, and

hence there exists some index at which the minimum is attained. If supp v X S � H,

then φpvq � 8. Since X XRzS is spanning in matpµ{Sq it contains j such that v j � 0.

The above argument in reverse is a proof of the other implication. l

Example. We continue the example just before Proposition 4.2.2. Let φ be the linear

map Q38 Ñ Q8 defined by the point pφpe1q,φpe2q,φpe3qq � p3, 3,1q. This map is in

the cone CM , and hence µφ � tt1,3u, t2,3uu. Thus,

φpvq � p3d v1q ` p1d v3q � p2d v2q ` p3d v3q, when v PLµ, (4.11)

according to Proposition 4.10 above. Write Hi for the set of v PQ38 for which φpvq �

φpeiqdvi . ThenLµ � H1YH3 andLµ � H2YH3, by (4.11). ThusLµ � pH1XH2qYH3.


(∞, 0,∞)

(∞,∞, 0)

(0,∞,∞)

H3

H2 H1

Lµ

Figure 4.6: The sectors Hi of Q38 on which the tropical linear function φ attains its

minimum at coordinate i overlayed on the (dashed) tropical linear space Lµ.

The situation is depicted in Figure 4.6.

Proposition 4.2.2 is an essential step in proving the theorem below. The second

part of this theorem gives an alternative characterization of the tropical linear space

associated to the orthogonal dual of µ, by Proposition 4.1.9. Of course, one needs to

consider this tropical linear space to be a submodule of the space of linear functionals

on FR, through the map v ÞÑ φv , where

φvpwq � xv, wy �àiPRpvi dwiq, w P FR.

The characterization thus obtained is essentially that of Proposition 2.3 of [59].

For the first part we need a definition. The restriction map ρ : pFRq� Ñ L �

is a homomorphism of semimodules, and since pFRq� is finitely generated and L �

is residuated, the dual residual of ρ exists when ψ P pFRq� and ψpvq � 8 for all

v P L implies that ψpwq � 8 for all w P FR. This is exactly the case when µ has no

loops. Then for all φ P L � the image ρ^pφq exists and is referred to as the minimalrepresentative of φ. Note that,

ρ^pφq � inftψ P pFRq� | φpvq ¤ψpvq for all v PL u. (4.12)

Note that since ρ is surjective, by abstract residuation theory one gets that the com-

position ρ^ �ρ : pFRq�Ñ pFRq� is a dual closure mapping (cf. [8, Theorem 2.7]), i.e.

it is increasing, idempotent and at most the identity.

Theorem 4.2.3 (i) the functional φ is the minimal representative of its restriction toLµ if and only µφ has no loops, (ii) the functional φ vanishes tropically on Lµ if and


only if µφ has no isthmuses, other than those of µ and φpe jq � 8 for all isthmuses j ofµ.

Proof. (i) For the first statement it is sufficient to prove that j P R is a non-loop

(i.e. t ju is independent) in µφ if and only if there is v P Lµ � p8, . . . ,8q such that

φpvq � φpe jq d v j and v j � 8. Suppose that j is independent. There exists some

basis B that contains j. Since B � j is not a spanning set, there must exist some

v P Lµ � p8, . . . ,8q such that for all i P B � j either φpvq φpeiq d vi or vi � 8.

Since B is a spanning set, for this v both φpvq � φpe jq d v j and v j �8.

Let v P Lµ � p8, . . . ,8q such that φpvq � φpe jq d v j and v j � 8. Denote the

support of φ by S. We distinguish two cases:

If j P S then the statement holds with v replaced by πpvq, where π : FR Ñ FS is

the canonical projection. Hence we can assume without loss of generality that R� S.

There exists a fundamental generator vI for which φpvIq � φpe jq d µpI � jq and

j R clpIq, by Lemma 4.1.4. Let B be a basis of µφ . By the assumption that R � S the

latter matroid equals respφ �µq. If j P B then j is independent and there is nothing to

prove, so suppose that j R B. By the exchange condition for φ �µ there exists i� P BzIsuch that,

pφ �µqpI � jq d pφ �µqpBq ¥ pφ �µqpI � i�q d pφ �µqpB� i�� jq.

Since pφ �µqpI � jq ¤ pφ �µqpI � i�q and neither equals 8 it follows that pφ �µqpB�i�� jq ¤ pφ �µqpBq. Thus B� i�� j is a basis of µφ and hence j is independent in µφ .

If j R S then φpvq � 8 and by linearity there exists some fundamental generator

vI whose j-th coordinate is not equal to 8 satisfying φpvIq � 8. Thus, clpIq � S and

j R clpIq, by Lemma 4.1.4. Thus j R clpSq and hence j is independent in matpµ{Sq.

(ii) Suppose that there exists v PLµ�p8, . . . ,8q such that φpvq � φpe jqd v j and

v j �8, and for all i P R� j either φpvq φpeiq d vi or vi �8. Then by Proposition

4.2.2 this is equivalent to the statement that all spanning sets of µφ contain j. This

characterizes j as an isthmus of µφ . Ifφ vanishes on v thenφpvq � 8. Hence,φpe jq �

8 and vi �8 for all i � j. Thus, R� j is a flat and j is an isthmus of µ.

(iii) Suppose that j is an isthmus of µφ that is not also an isthmus of µ. Then

R is a spanning set of µφ , while R � j is not and thus, by Proposition 4.2.2, there

exists v P Lµ such that φpvq � φpe jq d v j and v j � 8. Moreover, for all i � j either

φpe jq d v j φpeiq d vi or vi � 8. Clearly, φ doesn’t vanish tropically at v, unless

φpvq � 8. Suppose, for the sake of contradiction, that φpvq � 8. Then for all i � j it

holds that vi �8. Thus, R� j is a flat of µ and hence j is an isthmus of µ, contradicting

the assumption. Hence φ does not vanish at v.

Suppose that φ does not vanish tropically on Lµ and that every isthmus of µφ is

also an isthmus of µ. We show that there exists an isthmus j of µ for which φpe jq � 8.


There exists v P Lµ such that φpvq � φpe jq d v j � 8 and φpvq φpeiq d vi for all

i � j. By Proposition 4.2.2 this means that j is an isthmus of µφ and thus, by the

premise, that j is an isthmus of µ. But clearly, φpe jq � 8. l

Remark. In the case that F � B, that is, in the case of ordinary matroids the above

statements specialize to known facts. Let µ be an ordinary matroid on R. Recall that

Lµ � BR is isomorphic, as a lattice, to the lattice of flats of µ (cf. Proposition 4.1.5).

A linear function φ : BR Ñ B is defined by its support S. The matroid µφ equal µ{S`µrSs, where the operations are the well-known operations of restriction, contraction

and direct sum on ordinary matroids. Since µφ has no loops precisely when S is a flat

of µ it is a consequence of Theorem 4.2.3 that the linear functionsLµÑ B correspond

bijectively to the flats of µ.

By the last remark the lattices L and L � are anti-isomorphic in case F � B.

We cannot expect this result to be true for more general semifields, since when F is

infinite L has no minimum, whereas it has a unique maximum. However, a similar

result holds. To prove it we make use of some of the results in Section 4 of [13] that

we restated in Section 3.2 of this thesis.

Theorem 4.2.4 Let L be a tropical linear subspace of FR and denote the canonicalinclusion by ι,

(i) if F � B then the composition of the map BR Ñ pBRq� that assigns the linearfunctional defined by ei ÞÑ 0 if vi � 8 and ei ÞÑ 8 if vi � 0 to v P BR with ι� isan anti-isomorphism of lattices when restricted to L .

(ii) if F is infinite and L X pF�qR � H, then the composition of the map pF�qR ÑHomppF�qR, F�q that assigns the linear functional defined by ei ÞÑ v�1

i to v PpF�qR with ι� is an anti-isomorphism of lattices when restricted to L X pF�qR.

Proof. Let µ be a valuated matroid that has L as its tropical linear space. Denote the

matrix whose rows are the vectors vH , where H runs over set of hyperplanesH of µ,

by M . Then the row space of M is L , while the column space is t�φpvHq

�HPH | φ P

L �u. The latter is isomorphic toL � as an F -module. Part (i) now follows immediately

from Proposition 3.2.3. For part (ii) it is sufficient to prove that the isomorphism

between FH dM and Md FR restricts to an isomorphism between pFH dMqXpF�qR

and pM d FRq X pF�qH and is of the form as stated in the theorem.

Let v P pFH dMq X pF�qR. The image of v under the isomorphism of Proposition

3.2.3 is M dw, where w � inftw1 | xw1, vy ¥ 0u � v�1. Since L XpF�qR is not empty

µ has no loops. Thus, for each i P R there is H P H such that i R H and all of the

rows of M are different from p8, . . . ,8q. Hence, M d w P pF�qH . This proves that

the isomorphism indeed restricts as stated. Note that the inverse satisfies the same

property by symmetry. l


Remark. The result above is a consequence of Theorem 23 of [19] which gives a

particular map that establishes that the tropical convex sets generated by the rows

and column of a matrix over the finite tropical real numbers are isomorphic polyhedral

complexes. This result is very much related to the that of [13], which we refer to in

this thesis. Interpreted in the spirit of [19] a tropical linear space is the row space

of the matrix A whose rows are the canonical generators vH . The space of linear

functionals, defined by their images on vH , is the tropical linear subspace of FH that

is the column space of A.

The theorem gives a simpler way of recognizing which linear functionals are min-

imal representatives of their restrictions to a tropical linear subspace, at least when

the functional is finite on the finite part of FR.

Corollary 4.2.5 Suppose F is infinite and φ : pF�qR Ñ F� is linear. Then φ is aminimal representative of its restriction to L if and only if φpeiq

�1iPR PL .

Proof. This follows immediately from the proof of the theorem. However, it is also

a consequence of Theorem 4.2.3 once one realizes that µφ � pµKqKz , where z �

�pφpeiqqiPR. l

Remark. It follows from Theorem 4.2.4 that L � is not a tropical linear space in

general. In fact, for the case that F � B, the statement thatL � BR is a tropical linear

space is equivalent to the statement that it is a geometric lattice. Since it is known

that the opposite lattice of a geometric lattice is not necessarily geometric L � is not

in general a tropical linear space (cf. [12]).

The next statement is quite remarkable. Since it holds for both a valuated matroid

and its underlying matroid it states that the tropical rank of the matrix of generating

vectors equals the rank of the same matrix with every finite entry replaced by 0.

Proposition 4.2.6 LetL � FR be a tropical linear space. The space of linear functionalsL � is finitely generated by the restrictions of αi : FR Ñ F for i P R, where αipe jq � 0

if i � j and 8 otherwise. Let H denote the collection of hyperplanes of µ. The tropicalrank of the set of vectors t

�αipvHq

�HPH | i P Su equals the rank of S in µ.

Proof. The first statement follows immediately from the surjectivity of the restriction

map. Let S � R and write M for the matrix whose rows are the vectors in the state-

ment. We need to show that the largest tropically non-singular square submatrix of

M has size rkµpSq. Clearly, the row space of M is the image of L under the canonical

projection π : FR Ñ FS and the rank of S is the rank of the matroid µrSs. By Lemma

4.1.11 and Lemma 4.1.15 the row space of M generates LµrSs. It is thus sufficient to

prove the statement for S � R.


Let B be a basis of µ. Then M has |B| rows such that each is proportional to

precisely one of vB�i , for i P B. The submatrix of M with these rows and columns

indexed by the elements of B is 8 everywhere, except on the diagonal, where the

entries are elements of F�. This submatrix is clearly tropically non-singular. Thus, the

tropical rank of M ¥ |B| � rkµ � rkµpRq. Let M 1 be a square, tropically non-singular

submatrix of M . Suppose its columns are indexed by some set S � R. Then the tropical

rank of the full submatrix of M with columns indexed by S is |S|. In particular M has

at least |S| distinct rows that form a minimal set of generators of its row space. Since

the row space is precisely LµrSs it must be that these generators are of the form vH ,

for H a hyperplane of µrSs, by Lemma 4.1.4. Thus, matpµrSsq is a matroid on S with

|S| distinct hyperplanes satisfying the additional condition that there exists exactly

one bijection σ : H pµrSsq Ñ R such that σpHq R H, for all H P H pµrSsq. We show

that this implies that matpµrSsq is the unique matroid of rank |S| on S.

First, it is more convenient to prove the dual statement: if there exists precisely

one bijection σ between the circuits C and the ground set R of a matroid M such

that σpCq P C , then M is the rank 0 matroid on R. Suppose that this is not the case.

There exists p0 P R such that p0 is not a loop in M . Let C0 be the preimage of p0,

i.e. σpC0q � p0. Then C0 � p0 � H. Let p1 P C0 � p0 and let C1 be the preimage of

p1 under σ. By minimality of C0 among dependent sets, p1 is not a loop either. Since

p1 P C0 and σpp1q � σpp0q it follows that p1 is also not a loop. Repeating this process,

one obtains sequences pp0, p1, p2, . . .q and pC0, C1, C2, . . .q such that

σpCiq � pi , pi � pi�1 and pi�1 P Ci X Ci�1, for all i � 0, 1,2, . . ..

Since R is finite, there exist minimal indices k l such that Ck � Cl . Now construct a

new bijection σ1 : C Ñ R that equals σ on C ztCk, Ck�1, . . . , Cl�1u and has σ1pCiq �

pi�1 for all i � k, . . . , l � 1. Moreover, σ1pCq P C for all C P C . Thus, we have found

another bijection, contradicting the premise.

Since the circuits of the dual are the complements of the hyperplanes of the primal

matroid, we have proved the statement and hence matpµrSsq is the matroid of rank

|S| on S. Thus, |S| ¤ rkµ and hence the tropical rank of M is rkµ. l

Remark. The last part of the proof of the theorem also follows by the characterization

of the rank of a t0,8u-matrix given by Proposition 4.3. of [18].

Corollary 4.2.7 The tropical rank of the set of vectors tvH | H PH u equals the rank ofµ.

The above corollary has an important consequence for tropical linear spaces over

RYt8u. By Theorem 4.2 of [19] the dimension of a maximal face of the polyhedral

complex structure on the tropical linear space induced by the type function on points


equals the rank of an associated matroid. Note that although the results of [19] were

proved for tropical convex sets in Rn they hold for finitely generated submodules of

Rn8.

4.2.2 Extensions of matroids

Elementary extensions of matroids

This section deals with functions on tropical linear spaces that are restriction of trop-

ical polynomial functions. Let α : L Ñ F be such a function. In case that F � R8 a

polyhedral structure onL defines a polyhedral structure on the graph of α. However,

the resulting complex is in general not balanced. It is a result, proved for example in

[57] that one can always add maximal dimensional weighted faces to the complex

to make the result balanced. Moreover, this extension is unique and we refer to it

as the extended graph of α. Note that it is not so clear how to define the extended

graph intrinsically for the finite tropical semifield and therefore, in the first half of

this section we restrict ourselves to the case F � R8. We state a specialization of the

result exactly as we need it, but take some liberty with the terminology of polyhedral

complexes. For purposes of this section, a polyhedral complex in Rn8 is the collection

of topological closures of polyhedra in Rn that together form a polyhedral complex.

Lemma 4.2.8 Let L be a tropical linear subspace over RR8 and α : L Ñ R8 the

restriction of a tropical rational function to L . There is a unique balanced polyhedralcomplex whose underlying set is of the form

tpv,αpvqq | v PL uY tpv,λq | v PK and λ¥ αpvqu � RR8�R8, (4.13)

for some K � L , and whose polyhedral structure is induced by the refinement of thedomains of linearity of α on L and the polyhedral structure on L . Moreover, K doesnot depend on the polyhedral structure on L .

Since underlying sets of extended graphs do not depend on the polyhedral struc-

ture on L the notion of the extended graph of a function on a tropical linear space

makes sense. The extended graph then no longer comes with a polyhedral structure.

For our purposes we are mostly interested in extended graphs that are tropical

linear spaces. In particular, the following two questions are answered in the next

theorem: (i) for which functions α : L Ñ F is the extended graph a tropical linear

space, (ii) which tropical linear spaces are extended graphs of tropical polynomial

functions on L . We start by describing the setup on the matroid side.

Let µ be a matroid on R over F . A matroid ν on S over F is called an extension of

µ if: (i) R � S, (ii) rkµ� rkν , and (iii) νrRs � µ. If |SzR| � 1, then ν is said to be an


elementary extension of µ. Note that in the case that F � B these terms coincides with

standard matroid terminology (cf. Section 7.2. of [50]).

Let ν be an elementary extension of µ. Write S � R\ tpu and set d � rkµ. By

the second and third condition the values of ν on d-size subsets of S are fixed by µ.

Thus, the elementary extensions of µ coincide with a certain subset of all functions

Pd�1pSq Ñ F . The map establishing the coincidence assigns to any elementary ex-

tension ν the function I ÞÑ νpI \ tpuq. We refer to the functions in the image of this

map as elementary matroidal extension functions of µ, or simply, matroidal extensionfunctions.

Before we state the next theorem we remind the reader that pv : wq � inftλ P F |v ¤ λdwu, for v, w P FR and w � p8, . . . ,8q.

Theorem 4.2.9 Let µ be be a matroid on R over F of rank d. Then α :Pd�1pRq Ñ F isan elementary extension function of µ if and only if,

(i) αpIq � 8 if rk I d � 1, and(ii) αpIq � pvI : vJqdαpJq for all size d � 1 independent sets I and J that have equal

closure in µ, and(iii) for all S � R and i, j, k P R satisfying |S| � d � 2 and i, j, k R S the expression,

�µpSi jq dαpSkq

�`�µpSikq dαpS jq

�`�µpSk jq dαpSiq

�(4.14)

vanishes tropically

Proof. Let ν be an elementary extension of µ and denote the associated elementary

extension function by α. Note the difference between the Greek letter ν , which stands

for the elementary extension, and the symbols vI , written with a minuscule Latin v,

that stand for the canonical generators of Lµ.

Let I � R be a dependent subset of size d�1. Then rkνpI�pq ¤ rkµpIq�1¤ d�1.

Thus, I � p is not a basis of ν , or equivalently, νpI � pq � 8.

Let I , J � R be independent sets of size d � 1 such that clµpIq � clµpJq. Let i PRz clµpIq. We prove that νpI�iqdνpJ�pq � νpJ�iqdνpI�pq. By part (iii) of Theorem

4.1.3 the sum of µpJ�i�p� jqdµpI� jq for j running over J�i�p vanishes tropically.

The terms with j R ti, pu are 8 and hence νpI � iq d νpJ � pq � νpJ � iq d νpI � pq.Since vI and vJ are proportional it holds that pvI : vJq � pνpI � iq : νpJ � iqq. The

statement follows, since pνpI � iq : νpJ � iqq � νpI � iq d νpJ � iq�1.

Let S � R and i, j, k P R as in part (iii) of the theorem. Setting X � SYti, j, ku and

Y � S� p and applying part (iii) of Theorem 4.1.3 again yields the statement.

Now let α be a function satisfying (i), (ii) and (iii) of the theorem. We need to

show that the map ν : PdpR� pq Ñ F defined by νpAq � µpAq if A � R and νpAq �αpA� pq if p P A is a valuated matroid. We use characterization (iv) of Theorem 4.1.3.


Let X , Y � R� p of satisfying |Y | � d � 1, |X | � d � 1 and |Y zX | � 3. Set S � X X Yand write ti, j, k, lu � X zY \ Y zX . We distinguish three cases.

(1) Suppose X , Y � R. Then the condition (4.2) is satisfied since ν restricts to µ

and µ is a valuated matroid.

(2) Suppose p P ti, j, k, lu. Then the vanishing condition (iii) of Theorem 4.1.3 for

ν translates immediately to (4.14).

(3) Suppose p P S and denote S�p by T . If T is dependent in µ, then by condition

(i) all terms of expression (4.2) equal8, and hence the expression vanishes tropically.

Suppose that T is independent. For ease of notation write αpT Y Zq by x 1Z . Then for

each m R T the expression,

µpT mi jq d x 1kl `µpT mikq d x 1jl `µpT mk jq d x 1i j ,

vanishes tropically, by the second item in this proof. Since T is independent, µpT YZq � ξpZq, where ξ is the contraction of µ by T (cf. Corollary 4.1.12). For ease of

notation we assume without loss of generality that ti, j, k, lu � t1,2, 3,4u and omit

the multiplication sign from our computations. It is now sufficient to prove that the

expression,

x 112 x 134 ` x 113 x 124 ` x 114 x 123

vanishes tropically, if the expressions,

ξp j34qx 112 ` ξp j24qx 113 ` ξp j23qx 114, ξp j34qx 112 ` ξp j14qx 123 ` ξp j34qx 124

ξp j24qx 113 ` ξp j14qx 123 ` ξp j12qx 134, ξp j23qx 114 ` ξp j13qx 124 ` ξp j12qx 134

vanish tropically for any j R r4s, where ξ is a valuated matroid of rank 3. As a final

substitution, set xZ � x 1Z d ξp jr4szZq for some fixed j. Now it is sufficient to show

that if all ofx12 ` x13 ` x14, x12 ` x23 ` x24

x13 ` x23 ` x34, x14 ` x24 ` x34

vanish tropically, then x12 x34 ` x13 x24 ` x14 x23 vanishes tropically.

Take two 2-sets having empty intersection, say 12 and 34, without loss of gener-

ality, and suppose the minimum is attained at x12 x34. Now we need to show that,

x12 x34 ¥ x13 x24 or x12 x34 ¥ x14 x23, (4.15)

given (leaving out the equality conditions coming from the tropical equations)

px12 ¥ x13 or x12 ¥ x14q and px12 ¥ x24 or x12 ¥ x23q and

px34 ¥ x13 or x34 ¥ x32q and px34 ¥ x24 or x34 ¥ x14q


Suppose that (4.15) is not true to arrive at a contradiction. Without loss of generality

we may assume that x12 ¥ x13 (otherwise apply the permutation p3, 4q). Then x34

x24 and hence x34 � x14. But then x12 x23 and thus x12 � x24. Then x34 x13 and

thus x34 � x23. Here we used that the assumption that the quadratic equation is not

satisfied. But then x13 ¡ x34 � x14 � x23 ¡ x12 � x24 ¥ x13. l

Remark. Consider the uniform matroid µ � U3r4s over F , defined by µpAq � 0 for

all size 3 subsets A of r4s. The proof of the statement that functions Pd�1pRq Ñ Fthat satisfy conditions (i), (ii) and particularly (iii) of the theorem reduced to proving

that the solutions of the tropical equations describing quotients of µ are valuated

matroid. Closer examination of the proof in fact shows that the conditions describing

elementary quotients, if non-trivial, imply the valuated exchange conditions.

Corollary 4.2.10 The elementary extension functions of µ form a module under thepointwise operations, that embeds into FH , where H is the collection of (matroidal)hyperplanes of µ.

Remark. It would be nice to have the statement that the induced order is a lattice

order. This is true for F � B, since any submodule of a free B-module of finite rank is

finitely generated. One can then use Proposition 3.1.6. However, finite generatedness

is not immediate for F infinite. Note that it is known that for F � B the induced order

is a lattice, by [44] (Weak Maps of Combinatorial Geometries).

Example. Consider the matroid U3p4q over F and denote its set of hyperplanes byH .

Then H � t12, 13,14, 23,24, 34u, where we take i j to mean ti, ju. Note that H is

also the collection of subsets of size 2 of r4s. By the theorem, a function α :H Ñ F is

a matroidal extension function of U3p4q if the expressions,

αp12q `αp13q `αp14q, αp12q `αp23q `αp24q

αp13q `αp23q `αp34q, αp14q `αp24q `αp34q. (4.16)

vanish tropically. The solution set is the tropical linear space associated to the cycle

matroid of the complete graph K4, which has affine diagram:

12

13

143424

23


This is not immediately obvious. It is clear that the linear space of MpK4q is contained

in the space of extensions, since the linear forms (4.16) come from the induced cyclesof K4, where induced cycles are those cycles that are also induced subgraphs. The

picture of K4 we have in mind is given below.

13 12

14

34

23

24

The linear space is cut out by all cycles of K4. However, by a theorem of Yuster and Yu

([71], Theorem 8) the linear forms associated to induced cycles form a tropical basis,

and hence already cut out the tropical linear space.

The tropical linear spaceLMpK4qis generated as a subsemimodule of FH by vectors

corresponding to its hyperplanes

H1 � t12,13, 14u, H2 � t12,23, 24u, H2 � t13, 23,34u, H4 � t14, 24,34u

and

H12 � t12,34u, H13 � t13,24u, H14 � t14, 23u.

The corresponding generating vectors are

v1 � p8,8,8, 0, 0, 0q, v12 � p8, 0, 0, 0, 0,8q,

v2 � p8, 0, 0,8,8, 0q, v13 � p0,8, 0, 0,8, 0q,

v3 � p0,8, 0,8, 0,8q, v14 � p0,0,8,8, 0, 0q,

v4 � p0, 0,8, 0,8,8q,

where the coordinates are in the order p12,13, 14,23, 24,34q.

Let L � FR be a tropical linear space and µ a valuated matroid associated to L .

Denote the canonical projection FR � F Ñ FR by πR. Then, each tropical linear space

K � FR � F of rank equal to d having πRpK q � L has an associated valuated ma-

troid ν that is an elementary extension of µ. After all, by Lemma 4.1.11 and Lemma

4.1.15, it follows that the restriction of any valuated matroid associated to K is pro-

portional to µ. Moreover, any elementary extension of µ gives rise to a tropical linear

space in FR � F of equal rank whose image under π equals L . This shows that there

is a bijective correspondence between such tropical linear spaces and elementary ma-

troidal extension functions. The next theorem describes how to obtain the extension

function from the tropical linear space, without constructing the elementary extension

matroid.


We shall write ExtppL q for the tropical linear subspaces of FR�p of rank equal

to that of L whose image under πR equals L . Similarly, we write Extppµq for the

module of elementary extension of µ, whose ground set is R� p.

Theorem 4.2.11 Let L � FR be a tropical linear space of rank d and µ a valuatedmatroid associated to L . The map that assigns to each K P ExtppL q the functionεK :Pd�1pRq Ñ F given by

εK pIq �

#πp

�π^K pvIq

�, if rkµ I � d � 1,

8, if rkµ I d � 1

where πK is the restriction of πR : FR�p Ñ FR to K , establishes a bijection betweenExtppL q and the elementary matroidal extension functions of µ. Moreover,

πppπ^K pvqq �

àIpv : vµI q d εK pIq,

for all v PL , where I runs over all size d � 1 subsets of R.

Proof. Let ν be an elementary extension of µ such that Lν �K and set εL :L Ñ Fto be the elementary extension function corresponding to the extension ν of µ. Since

εK pIq � νpI � pq � πppvνI q, it is sufficient to prove that

π^K pvq �à

Ipv : vµI q d vνI , v PL ,

where I runs over all size d � 1 subsets of R. Let v P L . By Theorem 3.1.8 it holds

that,

π^K pvq �à

Jpv : πK pv

νJ qq d vνJ , v PL ,

where J runs over all size d � 1 subsets of R� p. Since πKpvνJ q � vνJ if J � R and that

fact that v PL it holds that

v � πK�à

Ipv : vµI q d vνI

�, v PL ,

where I runs over all size d � 1 subsets of R. Now the claim follows from πKppv :

vJq d vJq ¥ v and πpppv : vJq d vJq � 8 for when p P J and |J | � d � 1. l

Remark. For general F the notion of an elementary matroidal extension function is

analogous to the notion of a linear subclass for ordinary matroids (cf. [50, Section

7.2]). More precisely, if F � B and ε is an elementary matroidal extension function,

the set tclpIq | εpIq � 8u is a linear subclass. The inverse construction associates the

elementary matroidal extension function I ÞÑ 8 if and only if clpIq P X to a linear


subclass X �H .

In a similar vein, the function πp � π^K : L Ñ F of the above theorem is the

analogue of a modular cut.

Example. We continue the example of MpK4q with F � Q8. Consider the extension

matroid associated to the vector v12 � p8, 0, 0, 0, 0,8q, or equivalently, to the exten-

sion function ε defined by,

εpt1, 2uq � εpt3,4uq � 8 and εpt1, 3uq � εpt1,4uq � εpt2,3u � εpt2, 4uq � 0.

We construct the function πp �π^K . By Theorem 4.2.11 it equals,

w ÞÑ pw : v13q ` pw : v14q ` pw : v23q ` pw : v24q

� pw2 _w4q ` pw2 _w3q ` pw1 _w4q ` pw1 _w3q,

where wi _ w j � maxtwi , w ju. Note that wi _ w j equals pwi d w jq � pwi ` w jq on

tw | wi �8 or w j �8u. The latter expression is the tropical quotient of two tropical

polynomial functions.

By Theorem 4.2.11 the elementary matroidal extension functions and functions of

the form πp �π^K for K P ExtppL q are in bijective correspondence. We shall use the

name elementary matroidal extension function for the latter class of functions as well.

This is unlikely to lead to confusion, since the domain of a function is included in its

definition, or, at least, clear from the context.

Proposition 4.2.12 Let L � FR be a tropical linear space and Lν P ExtppL q. Denotethe associated matroidal extension function L Ñ F by ε. Then,

Lν � tpv,εpvqq | v PL uY tpv,λq | v PLν{p and λ¥ εpvqu.

When defined (e.g. when F � R8,Q8) Lν is the extended graph of ε.

Proof. We start by proving the easier inclusion �. By Theorem 4.2.11 any point in

FR�p of the form pv,εpvqq, with v P L is in K . Moreover, by Lemma 4.1.11 the

set tpv,8q | v P Lν{pu � K . For v P Lν{p and λ P F such λ ¥ εpvq the equality

pv,λq � pλ : εpvqqdpv,εpvqq`pv,8q holds. SinceK is closed under tropical addition

and scalar multiplication, the inclusion follows.

For every circuit C of ν choose a size d � 1-spanning set S of ν that contains Csuch that αCpeiq � νpS � iq. For the purpose of the proof we say such S represents

C . Note that by Corollary 4.1.10 we have that K is the intersection of the tropical

vanishing loci of functions of the form αC .


Let v P K . Then πK pvq P L , since K is an extension of L . We write π for πK .

By definition of ε the inequality vp ¥ εpvq holds. Suppose πpvq R Lν{p. Then there

exists a circuit C of ν{p such that πpvq is not in the vanishing locus of αν{pC . Let S � Rrepresent C and consider the expression,

àiPS�p

νpS� p� iq d vi ��à

iPSpν{pqpS� iq d vi

`�νpSq d vp

�,

where we have used that tpu is independent in ν . If this were not the case, then

K �L�t8u and the theorem holds. The first sum of the right hand side of the above

expression equals αν{pC pπpvqq, and since the entire expression vanishes tropically it

must hold that νpSq d vp � αν{pC pπpvqq. Thus, the fiber of π above πpvq is singleton.

Thus, v � pπpvq,εpvqq since pπpvq,εpvqq is contained in the fiber, by Theorem 4.2.11.

That K is the extended graph of ε is a consequence of the form of K as estab-

lished by the theorem and the fact that extended graphs are unique by Lemma 4.2.8.

l

The space Lν{p is a tropical linear subspace of L . It equals L if and only if

ε :L Ñ F is identically 8, but otherwise it is of codimension 1. One should see Lν{pas the locus of vanishing of ε. In the setting of tropical linear spaces there is no harm

in referring to it as the cycle of ε. See also [57] and [25].

Proposition 4.2.13 Linear functions L Ñ F are elementary extension functions. Anelementary extension function L Ñ F is the pointwise maximum of a finite number oflinear functions.

Proof. Let φ : L Ñ F be a linear extension. Then φ extends to a linear function

FR Ñ F . We denote the extension by φ as well. Set εpIq � φpvIq, for I a corank 1

independent subset of µ and εpIq � 8 otherwise. We first need to show that ε satisfies

the conditions of Theorem 4.14. Condition (i) is fulfilled by definition, and condition

(ii) is true by proportionality of canonical generators vI and vJ , when clpIq � clpJq,by Lemma 4.1.4. The third condition is a priori non-trivial. We prove an equivalent

statement, that is trivially stronger.

Let X , Y � R with |Y | � d�1 and |X | � d�2. We aim to show that the expression,

àiPY zX

µpY � iq d εpX � iq �à

iPY zX

µpY � iq dφpvX�iq (4.17)

�à

iPY zX

àjPRzpX�iq

φpe jq dµpY � iq dµpX � i� jq


vanishes tropically. Let i� P Y zX and j� P RzpX � i�q such that the minimum in the

above expression is attained for j � j� and i � i�. Then the above expression equals,

àiPY zX

φpe j�q dµpY � iq dµpX � j�� iq.

By the valuated exchange condition for µ the minimum is attained at i � i� and at

least one other index i. This shows that the minimum of (4.18) is attained at least

twice.

Let ε : L Ñ F be the extension function of the extension K of L and fix an

elementary extension ν of µ such that Lν � K . Let Sν denote the set of dependent

S � R such that S� p is a basis of ν . For each S P Sν denote the function,

w ÞÑ νpS� pq�1 d¸

iPS�p

νpS� iq dwi , w PL ,

by φS . We claim that ε � maxtφS | S P Sνu. For each v P L the values vp P F such

that pv, vpq PK are defined by the vanishing of the expressions,

� ¸iPS�p

νpS� pq�1 d νpS� iq d vi

` vp,

where S runs over the elements of Sν . Thus, εpvq �maxtφSpvq | S P Sνu. l

Corollary 4.2.14 Any elementary extension function of a valuated matroid of rank 2

induces a linear function on the associated tropical linear space, in the sense of Theorem4.2.11.

Proof. Let L be a tropical linear subspace of FR of rank 2 and let µ be a matroid

associated to L . Then a function ε :P1pRq Ñ F is an elementary matroidal extension

function if and only if the expressions,

µpi jq dαptku`µpikq dαpt juq`µp jkq dαptiuq

vanish tropically, for all distinct i, j, k P R. Comparing these expression with the ex-

pressions definingL as a prevariety yields that α is an elementary extension function

if and only if the vector�αptiuq

�P FR is an element of L .

Let Mµ denote the matrix whose pi, jq-th coefficient is the µpti, juq. The rows of

Mµ are the canonical generators of L and hence the row space of Mµ is precisely L .

Since a linear function is determined by its image on the canonical generators, and

every linear function onL is the restriction of a linear function FR Ñ F it follows that

the column space of Mµ is isomorphic to the module of linear functions on L , under


the isomorphism that maps such a function α : L Ñ F to the vector�αpvtiuq

�P FR.

Since Mµ is symmetric (this is the feature that distinguished the rank 2 case from

cases of higher rank) the image under this isomorphism is precisely L itself.

We have shown that both the space of linear functions and the space of elementary

extension functions are isomorphic to L , under the same isomorphism, when one

takes into account Theorem 4.2.11 and hence coincide. l

Proposition 4.2.15 Let µ be a valuated matroid on R over F which is realizable overthe valued field pK ,ωq, with ω : K Ñ F surjective. Then µ�α p is realizable over analgebraic extension of pK ,ωq when α is linear.

Proof. Denote the rank of µ by d and assume without loss of generality that R � rns.There exists a matrix M of size d� n with entries in K such that µpAq �ωpdet MrAsq,where MrAs denotes the full submatrix of M whose columns are indexed by the

elements in A. Let x1, . . . , xn be indeterminates and consider the linear function

f : pKdqn Ñ Krx1, . . . , xnsd defined by f pa1, . . . , anq � x1a1� . . .� xnan, with ai P Kd .

Denote the columns of M by mi , set m� f pm1, . . . , mnq and write M 1 for the d�n�1

matrix whose first n columns coincide with those of M and whose n� 1-th column is

m. For each A � rns of size d clearly ωpdet M 1rAsq � µpAq. Let A � rn� 1s of size dsuch that n� 1 P A. Then

detpMrA1sq �¸

iPrns

p�1qn�i x i detpMrA1� pn� 1q � isq

Assume without loss of generality that α : FR Ñ F . There exists an algebraic extension

of pL,ω1q of pK ,ωq and elements ai P L with ω1paiq � αpeiq such that for all A1 �rn� 1s of size d such that n� 1 P A,

ω1pdetpMrA1sq �àiPrns

ω1paiq dω1pdetpMrA1� pn� 1q � isqq

�àiPrns

αpeiq dµpA1� pn� 1q � iq

� αpvµA1�pn�1qq,

which proves the statement. l

Corollary 4.2.16 Let K be an algebraically closed field with surjective valuation ω :

K Ñ F and R a finite set. Then the image Gr2pKRq in its Plücker embedding under the

entry-wise application of ω equals Dr2pFRq in its realization of the space of proportional

equivalence classes of valuated matroids of rank 2.


The behavior of linear subspaces under arbitrary elementary extensions of the

containing linear space is complicated. Matters are somewhat simpler once restricted

to linear extensions.

Proposition 4.2.17 Let ι :Lµ �Lν be tropical linear spaces and φ :Lν Ñ F a linearfunctional. Then Lµ�ι�pφq p �Lν �φ p,

Proof. Denote the ground set of µ by R. Write d be the rank of µ and e for the rank of

ν . Denote µ�ι�pφq�p by µ1 and ν �φ p by ν 1. We need to show that for all I � R� pand S � R� p with |I | � d �1 and |S| � e�1 the function φS : FR�p Ñ F defined by

φSpeiq � ν1pS� iq if i P S and 8 otherwise, vanishes on vν

1

I .

We distinguish a number of cases. Write π for the canonical projection FR�p Ñ FR.

(i) Suppose p R S. Then π�pφSq vanishes on Lν . Since πpvIq P Lµ � Lν and

φSpepq � 8 it shows that φS vanishes on vI .

(ii) Suppose p P S and p R I . Then φS vanishes on Lν 1 and vI P Γφ1 � Γφ � Lν 1and hence φS vanishes on vI .

(iii) The most complicated case is when p P S and p P I . In effect we show that

πpvIq PLν 1{p. Here the linearity of φ is crucial. Consider the expression, where

we drop the superscripted matroids, and write wJ , respectively vJ for the fun-

damental generators of Lν , respectively Lµ,

φSpvIq �àiPSzI

ν 1pS� iq dµ1pI � iq

�àiPSzI

φpwS�i�pq dφpvI�i�pq

�àiPSzI

�àjφpe jqνpS� i� j� pq

d�à

k

φpekqµpI � i� k� pq

�àiPSzI

�àj,k

φpe jqφpekqνpS� i� j� pqµpI � i� k� pq

(4.18)

Suppose that the minimum in the above expression is attained in some triple

pi�, j�, k�q. Set S� � S � j�� p and I� � I � k�� p. There exists some i1 � i�

such that νpS� i�qµpI � i�q ¥ νpS� i1qµpI � i1q, since Lµ �Lν and hence the

minimum in 4.18 is also attained at pi1, j�, k�q. This shows that φSpvIq vanishes

tropically.

l

Example. As already stated, the behavior of general elementary extension functions

under restriction is much more complicated. In general such a restriction is not an

elementary extension function for the linear subspace and examples are not difficult

to construct.


Consider the so-called rigid line L in the tropical linear spaceM over B associ-

ated to the matroid U3pt1,2, 3,4uq. This is example [25]. Then L is the cycle ofMcut out by the extension function ε defined by ε : vi j ÞÑ 8 if and only if ti, ju � t1,2u

or t3, 4u, and 0 otherwise. The restriction of ε itself to L is not an extension func-

tion. After all, L � tv12, v34u � tp8,8, 0, 0q, p0, 0,0, 0q, p0,0,8,8q, p8,8,8,8qu

and εp0, 0,0, 0q � 0. In canonical coordinates,

p0, 0,0, 0q � 0d p8,8, 0, 0q ` 0d p0, 0,8,8q,

and thus ε does not satisfy Theorem 4.2.11.

Let H denote the collection of hyperplanes of µ and suppose that the injection

ι : Extpµq ãÑ FH of Corollary 4.2.10 is dually residuated (for example when F � Bor when Extpµq is finitely generated). The map γ� ι � ι^ is a closure operator whose

image is the module of extension functions Extpµq � FH . A possible construction that

presents itself naturally now is the following.

Let ν be a valuated matroid on R such that j : Lµ � Lν and let ε P Extpνq. Then

j�ε is not necessarily a extension function of µ, however γp j�εq is. Then construct

µ1 � µ�γp j�εq p. However, by the following example Lµ1 �Lν�εp.

This following is an example from Chapter 7 of [50] of two incompatible single

element extensions.

1

2

3

4

5

6

(a) the base matroid ν

1

2

3

4

5

6p

(b) the extension of the codi-mension one subspace of Lν

1

2

3

4

5

6q

(c) the matroid of the tropicallinear space Lν �φ q

Figure 4.7: Two non-linear extensions of a matroid µ

The contraction of Figure 4.7(b) by p is the rank two matroid with atoms (and

hyperplanes) t3u, t6u, t2,5u, t1,4u. We denote it by µ. The restriction j�φ of φ toLµmaps vµ3 ÞÑ 0, vµ6 ÞÑ 0, vµ25 ÞÑ 8 and vµ14 ÞÑ 8. This is not an extension function. The

closure γp j�φq is the function that is identically 8. This follows since all extension

functions on a rank 2 matroid are linear. And hence the extension µ �γp j�φq q is

µ`U0pqq, i.e. it adds q as a loop. The corresponding linear space is not a subspace of

Lν �φ q.


Linear extensions, directed networks and linking systems

This section aims to give a different interpretation of linear extensions of a matroid in

terms of matroids induced by graphs. Such an interpretation makes it easier to handle

a large of number of extensions, and in particular, leads the way to a better picture of

the tropical linear monoid, to be defined later.

The central concept in this section is that of a linking system, or rather of the

valuated analogue thereof. Linking systems were introduced in [56] to generalize

induction of a matroid through a directed network or a bipartite graph. Independently,

they came up in [42], in essence to abstract the patterns of singular minors of a

matrix. In this context there is an interesting link with the Bruhat decomposition of

the general linear group, which is expanded upon in the articles [46]. Finally, valuated

linking systems were introduced by Murota. We mention the articles [47] and [40].

The next definition is from [47]. The definition of a linking system is derived

from that of a matroid, and hence they are really nothing new. However, they form a

useful device to put certain results on valuated matroids that are induced from other

valuated matroids on directed networks in a nice context. Another reason to define

them here is that they are a useful vehicle to prove statements on the induction of

valuated matroids through directed networks and bipartite graphs, as we shall see

later.

Definition 4.2.1 (Linking system) Let R and S be finite sets. A function λ : P pRq �P pSq Ñ F is called a linking system on pR, Sq over F when the map µλ :P pR>Sq Ñ Rdefined by,

µλpX q � λpRzX , SX X q

is a valuated matroid on R > S over F satisfying µλpRq � 0. The map µλ is referred to

as the graph, or the representation matroid, of λ.

The name representation matroid comes from [42]. The next example of a class

of valuated matroids is the key to [42]. Over B linking systems abstract the patterns

of singularity and non-singularity of a matrix, in the same way that matroids abstract

the patterns of singularity of full submatrices (i.e. linear dependence).

Example. There is a notion of realizability for valuated linking systems, that reduces

to realizability of the representation matroid. Let K be a field equipped with some

valuation ω : K Ñ F and let A be a matrix over K whose rows and columns are

indexed by R, respectively S. Then the function that assigns to each equicardinal pair

pX , Y q of subsets X � R and Y � S the value ωpdetpArX , Y sq, where ArX , Y s denotes

the submatrix with rows indexed by X and columns indexed by Y , is a linking system.

Its representation matroid is realizable as well. The matrix that realizes it is pIR Aq,


where IR is the identity matrix, with entries in K , whose rows and columns are indexed

by R.

The exchange condition for valuated matroids translates easily to similar defining

conditions for valuated linking systems. For a proof of the equivalence we refer the

reader to [47]. We refer to R as the domain set or row set of and to S as the range setor column set of λ.

Proposition 4.2.18 A map λ :P pRq�P pSq Ñ F is a linking system if and only if bothof the following symmetrical vanishing conditions are satisfied.

(i) For each pX1, Y1q, pX2, Y2q PP pRq�P pSq and i1 P X1zX2 there is either j1 P Y1zY2

such that,

λpX1, Y1q dλpX2, Y2q ¥ λpX1 � i1, Y1 � j1q dλpX2 � i1, Y2 � j1q

or there is i2 P X2zX1 such that,

λpX1, Y1q dλpX2, Y2q ¥ λpX1 � i1 � i2, Y1q dλpX2 � i1 � i1, Y2q

(ii) For each pX1, Y1q, pX2, Y2q PP pRq�P pSq and j1 P Y1zY2 there is either i1 P X1zX2

such that,

λpX1, Y1q dλpX2, Y2q ¥ λpX1 � i1, Y1 � j1q dλpX2 � i1, Y2 � j1q

or there is j2 P Y2zY1 such that,

λpX1, Y1q dλpX2, Y2q ¥ λpX1, Y1 � j1 � j2q dλpX2, Y2 � j1 � j2q

Proof. See [47]. l

Constructions that produce linking systems from linking systems induce construc-

tions on the associated representation matroids. We define the cofactor linking sys-

tem and the restricted linking system, and prove, through the vehicle of the repre-

sentation matroid, that they indeed yield linking systems. These are the construc-

tions we need for what follows. We start by defining them on an arbitrary function

λ :P pRq �P pSq Ñ F .

(i) The cofactor system of λ, denoted λK, is the function P pRq�P pSq Ñ F defined

by λKpX , Y q � λpRzX , SzY q.(ii) Let pX , Y q P P pRq � P pSq. The restriction!of a linking system of λ to pA, Bq,

denoted λrA, Bs, is the function P pAq �P pBq Ñ F defined by λrA, BspX , Y q �λpX , Y q.

The next statement proves that these indeed define valuated linking systems and gives


the induced construction on the representation matroids.

Proposition 4.2.19 Let λ :P pRq �P pSq Ñ F be a function.

(i) The cofactor system λK is a linking system if and only if λ is a linking system andλpR, Sq � 8. Its representation matroid is the orthogonal dual of µλ.

(ii) If λ is a linking system and pA, Bq PP pRq�P pSq then λrA, Bs is a linking system.Its representation matroid is proportional to the representation matroid

�µλ � pAY

Sq�rAY Bs.

Proof. We prove the claim in order stated.

(i) Let λ be a linking system. Since the map λ ÞÑ λK is an involution, it is sufficient

to prove that λK is a linking system with representation matroid as stated. Let pX , Y q PP pRq �P pSq. Then,

λKpX , Y q � λpRzX , SzY q

� µλpX > pSzY qq

� µλppR > SqzpRzX > Y qq

� µKλ pRzX > Y q

Moreover, µKλ pRq � µλpSq � 8.

(ii) That λrA, Bs is a linking system follows immediately from Proposition 4.2.18.

The representation matroid of λrA, Bs is the matroid on A> B defined Z ÞÑ µλpRzZ >SX Zq, for Z � A> B. Consider the image of Z under µ1 �

�µ � pAY Sq

�rAY Bs,

µ1pZq � pµ � pAY SqqpZq

� µpZ Y pRzAqq

� µppRzZq Y pSX Zqq,

where the proportionality, denoted �, is a consequence of Lemma 4.1.11. l

The significance of linking systems is that they induce a matroid on the range set

from a matroid on the domain set. The construction is as follows. Let λ be a linking

system on pR, Sq over F . Let µ be a matroid over F on R. The image of µ under λ is the

map P pSq Ñ R given by, in its independent set characterization (cf. Section 4.1.2),

Y ÞÑàX�R

µpX q dλpX , Y q, Y � R. (4.19)

We denote this image by λpµq, by slight abuse of notation. The content of the next

statement is that λpµq is indeed a matroid.


Proposition 4.2.20 Let λ be a linking system over F on pR, Sq and µ a matroid over Fon R. Then λpµq is a matroid over F on S.

Proof. First note that without loss of generality, that there exists a basis A of µ and a

set B � S of equal cardinality such that λpA, Bq � 8. One can always truncate µ such

that this is the case.

We prove that the representation matroid of λpµq is pµλ Y µq � S. This is a fairly

simple computation. First note that the matroid µλYµ has R as a basis, and hence it

follows by Lemma 4.1.11 that the contraction to S is the map that assigns to Y � Sthe value of pµλYµqpY Y Rq. It is thus sufficient to show that

µλYµpY Y Rq �à

XλpX , Y q dµpX q,

where X runs over the subsets of R of cardinality |Y |. This is an easy computation

once on realizes that there exists a basis B of µ such that RzBY Y is a basis of µλ. l

By the above proposition linking systems function as maps on the set of all ma-

troids with ground set the domain set of the system. We pursue this analogy further.

Let κ and λ be linking systems on pR, Sq and pS, Tq. The composition of κ and λ is the

map P pRq �P pTq Ñ F defined by,

pX , Zq ÞÑàY�S

κpX , Y q dλpY, Zq.

We denote it by λ � κ. The next proposition states that this indeed defines a linking

system and that composition behaves naturally with respect to the action of a linking

system on sets of matroids.

Proposition 4.2.21 Let κ and λ be linking systems on pR, Sq and pS, Tq. Then the com-position λ � κ is a linking system on pR, Tq. Moreover, pλ � κqpµq � λpκpµqq for anymatroid µ on R.

Proof. The proof is similar to that of Proposition 4.2.20. We compute the graph of

λ �κ, as in Definition 4.2.1. Set γ� λ �κ and let A� R\ T . Then,

µγpAq � γpRzA, AX Tq

�àY�S

κpRzA, Y q dλpY, AX Tq

�àY�S

µκppAX Rq Y pSX Y qq dµλppSzY q Y pAX Tqq.

This is precisely the value of A in the matroid pµκ `µ0pTqq Y pµ0pRq `µλq{S, where

µ0pX q, for X � R, T denotes the valuated matroid of rank 0 on X that assigns 0 to H.


The proof of the second statement follows immediately by expanding both expressions

according to their definitions. l

In matroid theory linking systems as presented in [56] simultaneously generalize

a number of known constructions of matroids from other matroids and a bipartite

graph or network. In the valuated case the same results holds.

Let F be a semiring and R an arbitrary set. An F-weighted network on R is a directed

graph on R equipped with a weight function that assigns an element of F� to every

arc of the network. A node is a source node when it has no incoming arcs, and it is a

terminal node, or sink node when it has no outgoing edges.

A weighted directed network Γ induces a function on pairs of equicardinal subsets

of the nodes. Let pX , Y q be such a pair of subsets. A linking between X and Y is a

collection of node disjoint paths from a node in X to a node in Y . The weight of alinking is the tropical product (ordinary sum) of the weights of all paths contained in

it, where the weight of the path is the tropical product of the weights of the edges that

make up the path. The function induced from Γ, denoted λΓ, assigns to each pX , Y qthe weight of a linking between X and Y of minimal weight. In other words, it assigns

the tropical sum over all linkings between X and Y to pX , Y q.

The construction is similar for bipartite graphs. Let ∆ be a bipartite graph with

color classes R and S with edge weights in F . To each pair pX , Y q with X � R and

Y � S and of equal cardinality, the function λ∆ assigns the minimal weight of a

matching between X and Y .

Both constructions are intimately related. The precise statement is that a linking

systems induced by a directed network is the cofactor linking system of one induced

by a bipartite graph.

Proposition 4.2.22 Let λ : P pRq2 Ñ F be a function. Then λ is the weight functionof an F-weighted directed network on R if and only if λK is the weight function of anF-weighted bipartite graph on pR, Rq.

Proof. Follows the proof of Lemma 2.4.3 of [50], which establishes a bijection be-

tween vertex disjoint path systems in matchings and networks in weighted graphs.

l

The above statement is also the means to prove that the functions induced by

directed networks are indeed linking systems.

Proposition 4.2.23 Let F be a tropical semifield. The following statements hold.

(i) The weight function of an F-weighted bipartite graph is a linking system.(ii) The weight function of an F-weighted directed network is a linking system.


Proof. By Lemma 4.2.19 and Proposition 4.2.22 it is sufficient to prove the first state-

ment. Let ∆ be a bipartite graph and let pX1, Y1q and pX2, Y2q be pairs whose image

under the induced function λ∆ are not 8. If they were, the exchange condition holds

trivially. Let M1 and M2 be matchings between pX1, Y1q and pX2, Y2q that realize the

minimal weight. Let i P X1zX2. There is a path in M1 YM2 that starts in i and ends in

either Y1 or in X2zX1.

Suppose the former and denote the end point by j. Then pX1 � i, Y1 � jq and

pX2 � i, Y2 � iq are matchings having the same combined edges as M1 and M2. Thus,

the sum of their weights are equal and hence λ∆pX1� i, Y1� iqdλ∆pX2� i, Y2� iq is

at most the combined weight of M1 and M2.

Now, suppose that the path p ends in j P X2zX1. Create two new matchings M 11

and M 12 between pX1 � i� j, Y1q and pX2 � i� j, Y2q by

M 11 � M1zpYte | e P pXM2u

M 12 � M2zpYte | e P pXM1u.

In effect, M 1i is obtained from Mi by flipping the edges in p. Clearly, the combined

weights of M 11 and M 1

2 is that of M1 and M2, and hence λ∆pX1 � i � j, Y1q dλ∆pX2 �

i� j, Y2q is smaller. l

Given linking systems κ and λ with the same domain set R, but disjoint range

sets S and T , over the same tropical semifield one constructs a new linking system on

pR, S > Tq by,

pX , Y q ÞÑàκpX1, Y X Sq dλpX2, Y X Tq,

where the sum ranges over the partitions pX1, X2q of X . Elementary linear extensions

are the result of applying a linking system of a very specific form to the base matroid.

This is the content of the next proposition.

To state it we require a simple construction. The identity linking system on R,

denoted idR is the linking system induced by the bipartite graph with color classes Rand a disjoint copy of R, and edges pi, iq with i P R all of weight 0. In other words,

idRpX , Y q � 0 if X � Y and 8 otherwise.

Proposition 4.2.24 Let λ be a linking system on pR, tpuq over R and µ a matroid on Rover R. Let idR be the identity linking system on R and set ν �

�idR �λ

�pµq. Then ν is a

linear extension of µ. Vice versa, any linear extension is obtained in this way.Moreover, idR �λ is induced by the weighted bipartite graph on pR, R� pq with edge

weight function,

pi, jq ÞÑ

$'&'%

0 if i, j P R and i � j8 if i, j P R and i � jλptiu, tpuq if j � p.


Proof. Let X � R� p be a set of size rkµ. If X � R then the image of X under λpµq is

the image of X under idRpµq which equals µpX q. On the other hand, if p P X , then,

pidR �λqpµqpX q �àiPX

idRpX � p, X � pq dλptiu, tpuqdµpX � p� iq

� φpvµX�pq,

where φ is the restriction to Lµ of the map FR Ñ F that is determined by ei ÞÑ

λptiu, tpuq.The second statement is a consequence of the same computation backwards,

where we set φ to be the restriction to Lµ of the map ei ÞÑ λptiu, tpuq. The state-

ment on the bipartite graph follows immediately by writing down the linking system

associated to the weighted bipartite graph given. l

Another way to describe the linking system in the proposition is by means of

weighted directed networks. This is more convenient for iterated linear extensions.

Consider pidR � λq as in the proposition. Let Γ denote the graph on R� p with arc

weights pi, jq ÞÑ 8 if j � p and pi, pq � λptiu, tpuq. It is obtained from the bipartite

graph in the lemma by contracting the edges pi, iq, with i P R, and orienting the

remaining edges towards p. It hence carries the same information as this bipartite

graph. Now a simple computation shows that λΓrR, R�ps � idR�λ. We have depicted

an example of both the bipartite graph and the directed network associated to a linear

function φ : F4 Ñ F in Figure 4.8.

φ1 φ2 φ3 φ4

1 2 3 4

1 2 3 4 p

(a) Bipartite graph through which a val-uated matroid induces its elementary ex-tension associated to φ.

φ1 φ2 φ3 φ4

1 2 3 4

p

(b) The simpler directed network associ-ated to the bipartite graph. The edges aredirected towards p.

Figure 4.8: Bipartite graph and network associated to the elementary extension asso-ciated to the linear function φ : F4 Ñ F .

Consider the sequence of elementary linear extensions,

µ� µ0φ1 // µ1

φ2 // . . . φm // µm . (4.20)

Repeating the procedure above allows one to represent µm as the image of µ under

a linking system that comes from a directed network. Denote the ground set of µi


by Ri and write tpiu � RizRi�1. The function φi is a linear map FRi Ñ F and hence

different choices of φi might restrict to the same function of Lµi�1. Build up the

directed network inductively in m steps and denote the weighted network obtained

after the i-th step by Γi .

(i) Γ0 is the network with node set R and no arcs,

(ii) Γi�1 is obtained from Γi by adding pi�1 to the node set and arcs p j, pi�1q with

weights φi�1pe jq to the arc set, for j in the node set of Γi .

Note that the node set of Γi is exactly Ri . The next theorem proves that the iterated

linear extension µm of µ is represented by a restriction of the linking systems induced

by Γm.

Theorem 4.2.25 Consider the sequence (4.20) of linear extensions and set Γ � Γm.Then µm � λΓrR, R Y tp1, . . . , pmuspµq. Moreover, λΓrR, R Y tp1, . . . , pmus � idR �

λΓrR, tp1, . . . , pmus.

Proof. The first statement is a consequence of Proposition 4.2.24. The last statement

follows since any vertex disjoint linking between pX , Y q must link YXR to YXR. Such

a linking only exists when Y X R� X . l

Example. This is an example for F � B, i.e. when the valuated matroids under consid-

eration are ordinary matroids. We remind the reader of the fact that a minimal linear

(functional) extension of a linear function Lν Ñ B is of the form φF for some flat Fof ν , where φF peiq � 0 if and only if i P F . Moreover, distinct flats are associated with

distinct linear extensions and the restriction of some φX to Lν equals the restriction

of φX to Lν , where X is the closure of X in ν .

1

2

3

4

5

6

7 8

(a) Iterated linear (principal) extension ofthe free matroid on t1,2, 3,4u

1 2 3 4

5

6

7

8

(b) A graph representing the iter-ated linear extension of (a)

Figure 4.9: Four-fold linear extension of a free matroid and graph Γ4


For an example, denote the matroid of the point configuration restricted to points

t1,2, . . . , iu in Figure 4.9 by µi , e.g. µ4 is the free matroid on 4 points. Then the

elementary extension µ6 of µ5 has the restriction of φt1,2,3,5u to Lµ5as extension

function. This restriction equals the restriction of φt1,2,3u, φt1,2,5u, φt1,3,5u, or φt2,3,5u

to Lν . Different choices of functional extension give rise to different linking systems

as in Proposition 4.2.24.

The graph associated to the canonical order of extension, i.e. µ4 Ñ µ5 Ñ µ6 Ñ

µ7 Ñ µ8 and minimal functional extensions of the matroidal extension functions is

depicted in Figure 4.9(b). Note that since we do not draw arcs of weight 8 the drawn

arcs necessarily have weight 0.

Proposition 4.2.26 Let λ be a linking system on pR, Sq. The image of the (trivial) ma-troid that assigns 0 to R under idR �λ is the representation matroid of λ.

Proof. The proof is a small computation. Denote the trivial matroid referred to in the

statement by µ and let Y � S be of size |R|. Then,

pidR �λqpµqpZq � idRpZ X R, Z X Rq dλpRzZ , Z X Sq dµpRq

� λpRzZ , Z X Sq

� µλ

��RzpRzZq

�\�Z X S

�� µλpZq,

which proves the statement. l

It is convenient to extend the action of linking systems on valuated matroids on the

appropriate ground set to tropical linear subspaces of an appropriate ambient affine

space. Formally, let λ be a linking system over F on pR, Sq and L � FR a tropical

linear space. Then the image of L under λ is defined to be the tropical linear space

associated to λpµq, where µ satisfies Lµ �L . Since the map on matroids induced by

λ is homogeneous, the image λpK q does not depend on the specific choice of µ.

Linking systems on pR, Sq over F fulfill the role of linear maps from FR to FS in

the sense that they map linear subspaces of FR (matroids on R) to linear subspaces on

FS (matroids on S). In the light of the previous theorem, the name graph for what is

called the representation matroid in [42] is appropriate in this setting. The situation

is more complex when we wish to restrict linking systems to proper subspaces of FR,

since in this case distinct linking systems on pR, Sq might represent the same map.

However, as in ordinary linear algebra, linear maps are represented by their graphs.

In the broader context of a linking system λ on pR, Sq restricted to a matroid µ on Rthe graph of the restriction of λ to µ is the matroid pidR � λqpµq. Linking systems λ

and κ, both on pR, Sq are said to define the same restriction to µ when their graphs

4.3. MORPHISMS BETWEEN LINEAR SPACES 113

pidR � λqpµq and pidR � κqpµq are equal. Note that when µ has rank |R| the graphs

are only equal when the linking systems are equal, by the above proposition and the

fact implicit in the definition that linking systems are uniquely determined by their

representation matroids.

4.3 Morphisms between linear spaces

In this section we use the results of Section 4.2.1 to construct the morphisms in a

category whose objects are tropical linear spaces. We let F denote a fixed semifield.

A morphism from L to M is a tropical linear subspace of L �M that is of the

form pidR �λqpL q, where λ is a linking system induced by iterated linear extensions

and a projection. The linking systems are fairly easy to handle, while the morphism

itself is not. Most of this section is devoted to proving that certain construction are

independent of the specific linking system that we chose to represent the morphism.

The central statement in the first part of this section (Proposition 4.3.3) is an example.

It states that restriction is well-defined. Unfortunately, we have not been able to prove

it. Its validity, and thus the validity of most of the theorems following it (including

Proposition 4.3.6, that states that composition is well defined) depends on Conjecture

4.3.1.

However, since a morphism with domain FR is uniquely defined by a linking sys-

tem (cf. Corollary 4.3.2), none of these problems occur when we consider only such

morphisms. In particular, HompFR, FRq has the structure of a monoid under com-

position. In Section 4.3.2 we define the tropical linear monoid as a submonoid of

HompFR, FRq and prove its most important properties in Theorem 4.3.7

4.3.1 A category of tropical linear spaces

The objects of the categoryB are the tropical linear spaces. By definition, a morphismα in B from a tropical linear space L to a tropical linear space M , denoted α :

L ÑB M , is a tropical linear subspace of L �M of the same rank as L , that is

the projection of an iterated linear extension of L . In other words, α P DrdpFR� FSq,

where d � rkL , L � FR andM � FS , is a morphism L ÑB M when there exists a

set T � R > S and a tropical linear space K � F T such that,

(i) there exist a filtration R � T0 � T1 � T2 � . . . � Tk � T , with |Ti�1zTi | � 1 and

a sequence of tropical linear spaces L �K0 �K1 �K2 � . . . �Kk �K such

that Ki � F Ti and Ki�1 is a linear extension of Ki ,

(ii) the image of K under the canonical projection to FR � FS equals α.

When confusion is unlikely to occur we drop the subscript B and write α : L ÑMinstead. In this section HompL ,M q denotes the set of all morphisms with domain L


and rangeM .

The next theorem is concerned with obtaining a representation of morphisms in

terms of linking systems induced by certain weighted acyclic networks. It has as an

important corollary that morphisms between affine spaces can be identified with such

linking systems.

Theorem 4.3.1 Let Γ be an F-weighted acyclic directed network with source vertices Rand sink vertices S having R X S � H. Let K � FR be a tropical linear space. ThenpidR �λΓrR, SsqpK q is a morphism K Ñ FS . Moreover, all morphisms in HompK , FSq

are of this form.

Proof. Denote the edge weight function of Γ by w. Consider the topological orderon the nodes N of Γ, i.e. p ¤1 q if there is a directed path in Γ from p to q. Fix an

arbitrary linear refinement ¤ of ¤1. Let p1 be the ¤-minimal element of NzR. Then

all arcs with end point p start in R. Let φ1 be the linear function FR Ñ F defined by

φpeiq � wpi, p1q. Add the rest of the nodes inductively. Finally one obtains a tropical

linear space in F N , which equals pidR � λrR, NzRsqpK q by Theorem 4.2.25, and is

an iterated linear extension of K . Finally, the projection of this space to R \ S is

pidR �λrR, SsqpK q P HompK , FSq.

If α P HompK , FSq. Then there exist T � R\ S and α1 P DrpF T q such that α1rR\Ss � α and α1 is an iterated linear extension of K , i.e.,

Kφ1 // K1

φ2 // K2φ3 // . . . φm // α1

Choose a matroid associated to K and construct the weighted acyclic network Γm

as in the paragraph preceding Theorem 4.2.25. By the preceding paragraph pidR �

λΓmrR, Ssq � α. l

Let L � FR andM � FS be tropical linear spaces. A linking system λ on pR, Sqover F is said to represent a morphism α P HompL ,M q when α � pidR � λqpL q. In

this terminology the previous theorem states that every morphism is represented by

the restriction of a linking system induced by a weighted directed acyclic network. It

is an immediate consequence of Proposition 4.2.26 that a linking system representing

a morphism with domain FR is unique. This is the content of the next statement.

Corollary 4.3.2 The map λΓrR, Ss ÞÑ pidR � λΓrR, SsqpFRq implicit in the theorem isbijective.

In essence, the corollary combined with the theorem state that any morphism

K Ñ L is a restriction of a morphism FR Ñ FS , where K � FR and L � FS . This

statement has a more general form, in the sense that a morphism on K restricts to a


morphism on any subspace. The proof is by representing a morphism as a linking sys-

tem. The main difficulty lies in proving that different representations yield the same

restriction. As a matter of fact, the proof of the statement depends on the following

conjecture, which we have not been able to prove.

Conjecture 4.3.1 Let R� S be finite sets and denote the canonical projection π : FR Ñ

FS by π. Consider K �L � FR and suppose that there exists an iterated extension,

σ : πpK q // . . . // K ,

such that the iterated extension τ : Lztpu Ñ L associated to the order on RzS corre-sponding to σ is the restriction. Then for any p P RzS the extension function LztpuÑLis the restriction of the extension function KztpuÑK .

Proposition 4.3.3 Suppose Conjecture 4.3.1 is true. Let α P HompK ,L q be a mor-phism, for K � FR and L � FS . Let ξ be a matroid such that Lξ � K . For λ and κlinking systems on pR, Sq representing the same morphism in HompK ,L q it holds thatpidR �λqpξq � pidR �κqpξq.

Proof. Let α be a morphism in HompK ,L q. Let FR and FS be the ambient spaces of

K and L . Assume without loss of generality that RX S �H. There exists a tropical

linear spaceM � F T , with R\ S � T , whose image under the canonical projection

π : F T Ñ FR � F T equals α. Moreover,M is an iterated linear extension of K �K0,

as follows,

K0φ1 // K1

φ2 // K2φ3 // . . . φm //M (4.21)

Write T � R \ tp1, . . . , pmu, where R \ tp1, . . . , piu is the index set of the ambient

affine space containing Ki . Fix a linear order on RzT by pi ¤ p j if and only i ¤ j.Write S � tq1, . . . , qnu � TzR, with qi ¤ q j for i ¤ j. The situation is summarized in

the next diagram.

K0φ1; p1 //

ε1; q1

!!BBB

BBBB

BK1

φ2; p2 // K2φ3; p3 // K3

φ4; p4 // K4φ5; p5 // . . . φm; pm //M

π

��L1

ε2; q2 // L2ε3; q3 // L3

ε4; q4 // L4ε5; q5 // . . . εn; qn // α,

(4.22)

in whichLi is the projection ofM to R\tq1, . . . , qiu and εi is defined by the diagram,

i.e. it is the unique extension function associated to the extension Li of Li�1.

Now letK 1 �K andK 1i �Ki , whereK 1

i is defined iteratively to be the extension

of K 1i�1 associated to φ1i , with φ1i the restriction of φi to Ki . Again, set L 1

i to be

the projection of K 1m to FR � Ftq1,...,qiu. Thus, L 1

i � Li and we have the diagram of


subspaces,

K 10

φ11; p1//

ε11; q1

AA

AAAA

AAK 1

1

φ12; p2// K 1

2

φ13; p3// K 1

3

φ14; p4// K 1

4

φ15; p5// . . .

φ1m; pm// K 1

m

π

��

L 11

ε12; q2// L 1

2

ε13; q3// L 1

3

ε14; q4// L 1

4

ε15; q5// . . .

ε1n; qn// L 1

n,

(4.23)

The aim is to prove that L 1n does not depend on the specific choice of iterated se-

quence, but only on α. Since α is uniquely determined by K and ε1, . . . ,εn it suffices

to prove that ε1i is the restriction of εi to L 1i�1. We then set L 1

n to be the restricted

morphism.

By induction on n we prove a slightly more general statement. Consider the dia-

grams (4.22) and (4.23) in which the φi are general matroidal extension functions

that restrict to matroidal extension functions φ1i on the K 1i�1 (as opposed to linear

functions). Then ε1i is the restriction of εi .

If n � 1, then without loss of generality we may assume that q1 � pm. We start

by examining (4.23). Write ρ1i for the dual residual of the canonical projectionK 1i Ñ

K 1i�1 and denote the dual residual of the canonical projection L 1

1 ÑK1

0 by ρ1. Note

that ρ1ipvq � pv,φ1ipvqq and ρ1pvq � pv,ε1pvqq, by Theorem 4.2.11. Write πi for the

canonical projection FR� Ftp1,...,piu� FtpmuÑ FR� Ftp1,...,pi�1u� Ftpmu. Then repeated

application of Lemma 4.3.1 yields that

ρ1 � π1 �π2 � . . . �πm�1 �ρ1m �ρ

1m�1 � . . . �ρ11.

The same procedure, applied to the containing sequence (4.22), gives that,

ρ � π1 �π2 � . . . �πm �ρm �ρm�1 � . . . �ρ1.

By the above expressions for ρ and ρ1, and the fact that ρ1i is the restriction of ρi it

follows that ρ1 is the restriction of ρ to K 10 .

Suppose that the statement is true for all integers smaller than n. By the previous

paragraph ε11 is the restriction of ε1 to K 10 . Let j � 1, . . . , m such that q1 � p j . The

restriction of M to R \ tp1, . . . , p ju. For each i ¥ j, denote by π� the canonical

projection of FR � Ftp1,...,piu to FR � Ftp j ,...,piu. Note that we use the same symbol to

denote different projections. Consider the diagram of elementary extensions,

π�pK jqψ j�1

//

ε2

%%LLLLLLLLLLLπ�pK j�1q

ψ j�2// . . .

ψm�1// π�pKm�1q

ψm�1// π�pM q

��L2

ε3 // . . .εn�1

// Ln�1εn // α,


obtained from (4.22) by applying π� to the upper row. Note that π�pK jq �L1, which

has become the base in the diagram to extend from.

Denote the extension from π�pK 1i�1q to π�pK 1

i q by ψ1i . If it is true that ψ1

i is the

restriction of ψi to π�pK 1i�1q we can use induction and conclude the truth of the

statement. Here we make use of Conjecture 4.3.1. l

Let L � FR andM � FS . Let α P HompL ,M q be a morphism and K a tropical

linear subspace of L . The restriction of α toK is the morphism pidR�λqpK q, where

λ is a linking system representing α. The previous proposition shows that restriction

is well-defined, if Conjecture 4.3.1 is true. Moreover, since linear extensions respect

subspaces, it is a morphism K Ñ M (as opposed to a morphism K Ñ FS). We

denote the restriction of α to K by α|K .

The proposition above also allows us to generalize the action of linking systems

on tropical linear spaces, defined at the end of Section 4.2.2. A natural way is to

represent the morphism by a linking system and have the linking system act on the

tropical linear space. It is a consequence of the previous proposition and the next

lemma that this is well-defined.

Lemma 4.3.4 Let L � FR be a tropical linear space and λ a linking system on pR, Sqover F. Then λpL q � pidR �λqpL qrSs.

Proof. Let µ be a valuated matroid on R over F associated to L . For Y � S,

pidR �λqpµqpY q �à

X�YXRλpX zY, Y X Sq dµpX q �

àXλpX , Y q dµpX q,

where X � R. This proves the statement. l

Let L � FR be a tropical linear space. Denote the collection of all valuated ma-

troids on R over F whose associated linear space is a subspace ofL by Matpµq, where

µ is a valuated matroid satisfying L � Lµ. By the characterization of valuated ma-

troids associated to a tropical linear space DrpL q is the projectivization of Drpµq. In

particular, the latter only depends on the proportional equivalence class of µ.

Given a tropical linear spaceM � Lν � FS and a morphism α : L ÑM define

a map Matpµq Ñ MatpSq by Matpαqpξq � λpξq, where λ is a linking system on pR, Sqrepresenting α. By Proposition 4.3.3 the image λpξq is independent of the choice of

representing linking system and hence Matpαq is well-defined.

Its most important properties are stated in the next proposition.

Proposition 4.3.5 Suppose Conjecture 4.3.1 is true. The map α ÞÑ Matpαq is injective.Moreover, the image of Matpαq is contained in Matpνq.

Proof. Let α P HompLµ,Mνq and ξ P Matpµq. Then by Proposition 4.3.3 the matroid

pidR � αqpµq is well defined. Since the tropical linear space associated to αpµq is


contained in α, by Proposition 4.1.11 the tropical linear space associated to αpµq �

pidR �αqpµqrSs is contained in Lν . Thus, αpµq PMatpνq.

We need to show that this map is injective. Let Y � R > S with |Y | � rkµ. Then,

�pidR �λqpµq

�pY q �

àX

pidR �λqpX , Y q dµpX q

�à

X�YXRλpX zY, Y X Sq dµpX q

�à

X�YXRλpX zY, Y X Sq dµppX zY q Y pY X Rqq,

where X runs over the basis of µ. In particular, if this expression is not equal to 8,

there exists a basis of µ containing Y XR, and hence Y XR is independent in µ. Thus,

the contraction�µ{pRX Y qqpZq � µpZ Y pRX Y qq (cf. Lemma 4.1.11). Thus,

�pidR �λqpµqpY qq �

àYXRPI pµq

λpµ{pRX Y q `υ0pRX Y qq,

with υ0pR X Y q the matroid of rank 0 on R X Y that maps H to 0. It remains to

show that ξ � µ{pR X Y q ` υ0pR X Y q is in Matpµq. This is a consequence of the

characterization of contraction and direct sum. The tropical linear space associated to

ξ is tv PLµ | vi �8 for i P clµpRX Y qu. l

By homogeneity of the map Matpαq it induces a map DrpL q Ñ DrpM q and hence

morphisms determine natural maps on subspaces, if Conjecture 4.3.1 is true. It is

clearly not the case that a morphism is determined by its map on subspaces and it

is not even evident that in general the morphism is determined up to tropical scalar

multiplication by its map on subspaces.

Example. A simple example to see that the morphism α is not determined by its

map on subspaces of L is given by HompFR, Fq. Such a morphism induces a map

DrpFRq Ñ DrpFq. The latter consists only of two points F and 8. Any morphism

HompFR, Fq is given by a single linear elementary extension of a free matroid on

R and hence by a linear function FR Ñ F . Two linear function that have the same

support, induce the same map DrpFRq Ñ DrpFq.

Example. We give a number of examples of the above map. Among other things it

serves to convince the reader that the map on points of a projectivized tropical linear

space (i.e. the map on rank 1 quotient matroids) is the projectivization of a linear

map, and that any such map can be obtained in this way.

(i) The first example is of the case F � B. Consider the free matroid µ on R� r3s,

i.e, µpRq � 0. The associated lattice of flats (or tropical linear space) is the power set

lattice of R. The projectivization of this lattice as a module is the poset RztHu, which


we represent as the points in the barycentric subdivision of the triangle, as drawn in

Figure 4.10.

(∞,∞, 0)

(0,∞,∞)(∞, 0,∞)

(0, 0, 0)

(∞, 0, 0) (0,∞, 0)

(0, 0,∞)

Figure 4.10: Poset isomorphic to the projectivization of the tropical linear spaceLµ �B3 under the map v ÞÑ Rz supp v.

Note that the image of p8,8,8q under any morphism with domain B3 is always

the point p8, . . . ,8q and hence no information on the morphism is truly lost by pro-

jectivizing (this is peculiar to the case F � B). Let |S| � 3. A morphism α : BR Ñ BS

(or, with slight abuse of notation, a morphism B3 Ñ B3) defines a map on the points

of PpBRq (of which there are 7), and a map on the lines of PpB3q (of which there are 7

as well). Consider for example the morphism represented by network in Figure 4.11

and its induced map on subspaces of B3 of projective dimension 0 and 1.

7→

7→

7→

7→

7→

7→

7→

7→

7→

7→

7→

7→

7→

7→

λ ∼

Figure 4.11: The map induced by λ on the tropical linear subspaces of B3 of projectivedimension (rank) 0 and 1. It maps B3 to B3 (of projective dimension 2). The linesconsist of the points they contain.

The homogeneous map on points PpB3q Ñ PpB3q induced by λ equals pv1 : v2 :

v3q ÞÑ pv1 ` v3 : v2 : v1q. This is easily seen by realizing that such a projective point

has associated matroid µ : t1u ÞÑ v1, t2u ÞÑ v2, t3u ÞÑ v3 and thus that the action of λ


on such a point is

t1u ÞÑ λpt1u, t1uqdµpt1uq`λpt3u, t1uqdµpt3uq � v1 ` v3,

t2u ÞÑ λpt2u, t2uqdµpt2uq � v2,

t3u ÞÑ λpt1u, t3uqdµpt1uq � v1.

(ii) Example of the projective line over Q8. We aim to clarify figures 4.1(a) and

4.1(b). Morphisms with domain a tropical linear space of rank 2 are much easier to

understand than general morphisms, for the same reason that valuated matroids of

rank 2 are easier to understand. The essential difference between the rank 2 case and

the higher rank cases is that any elementary extension of a rank 2 valuated matroid

is linear. In particular, the set of morphisms HompL ,M q, with L of rank 2 are all

tropical linear subspaces of L �M of rank 2 that project surjectively onto L . We

give an example with L � F2.

The elementary extensions of F2 correspond bijectively to linear functions

px , yq ÞÑ pad xq ` pbd yq.

It remains to define composition of morphisms inB . To do so the next proposition

is required. First we state a construction that is required in the proof. Let R1, R2, S1, S2

be finite sets, which, without loss of generality, we can assume to be disjoint. For

linking systems κ on pR1, S1q and λ on pR2, S2q define the linking system pκ,λq on

pR1 > R2, S1 > S2q by the expression,

pκ,λqpX , Y q � κpX X R1, Y X S1q dλpX X R2, Y X S2q, .

for all X � R1 > R2 and Y � S1 > S2. This is easily seen to define a linking system,

either by using the definition (the representation matroid is the direct sum of the rep-

resentation matroids), or the exchange conditions for linking system (cf. Proposition

4.2.18). We denote it by pκ,λq.

Proposition 4.3.6 Suppose that Conjecture 4.3.1 is true. Let α P HompK ,L q and β PHompL ,M q. For any choice of representing linking systems λ for α and κ for β , thecomposition κ �λ represents the same morphism in HompK ,M q.

Proof. The proof relies on a decomposition lemma for the linking system pidR�pλ�κqq.

We prove that pidR�λ�κq � pidR,λq�pidR�κq. Let X � R and Z � R>T and consider


the image of pX , Zq under the composition of idR �κ and pidR,λq,

pX , Zq ÞÑà

Y�R>Spid�κqpX , Y q d pid,λqpY, Zq

�à

Y�R>SYXR�X

κpX zY, Y X Sq d idRpY X R, Z X Rq dλpY X S, Z X Tq

�à

Y 1�S

κpX zY, Y 1q dλpY 1, Z X Tq

� pλ �κqpX zZ , Z X Tq

� idR � pλ �κqpX , Zq.

This proves the decomposition.

It remains to prove that the tropical linear space pidR � pλ �κqqpK q � pidR,λqpαq

depends only on α � pidR � κqpK q and β � pidS �λqpL q. Let X � R and Z � T and

compute,

pidR,λqpαqpX > Zq �àY�S

αpX Y Y q dλpY, Zq

Now Y 1 ÞÑ αpX YpY 1XSqq is a valuated matroid (it is α{X ) if X is independent in the

matroid associated to α (or equivalently, in the matroid associated to K ). Its tropical

linear space is contained inL , since it is the projection of α{X`υ0pRq, whose tropical

linear space is contained in α (by the same argument as used in Proposition 4.3.5).

Now,

pidR,λqpαqpX > Zq �à

XPI pµq

�λpα{X q

�pZq.

By Proposition 4.3.5 the summands are determined uniquely by the morphism β , and

hence pidR,λqpαq is determined uniquely by β . This concludes the proof. l

The composition of α and β in the proposition is the morphism in HompK ,M q

represented by κ�λ. Note that in fact the proposition states that the restriction of the

composition of appropriate extensions of morphisms does not in fact depend on the

specific extensions. This is a natural and somewhat expected condition.

4.3.2 The tropical linear monoid

Definition and basic results

This section specializes the result of the previous section to morphisms HompFR, FRq.

It ends with a conjecture on the structure of HompFR, FRq that depends on the graphs

representing the linking system (and hence the morphism, since morphisms are rep-

resented uniquely by linking systems, cf. Corollary 4.3.2.

The set HompFR, FRq carries a composition which makes it into a monoid with


identity the morphism represented by idR. Moreover, the results which depend on

Conjecture 4.3.1, hold without this dependence in HompFR, FRq (or in fact for any set

of morphisms whose domain is a tropical affine space), since in that case morphisms

are uniquely represented by linking systems and there is no need to prove the inde-

pendence of maps on quotients from the representing linking system. We will exploit

the latter fact by mostly working with the linking systems themselves, as opposed to

the morphisms.

Let F be a tropical semifield and n P N. The tropical linear monoid of rank n over Fis the submonoid of HompF n, F nq consisting of the morphisms (linking systems) that

map F n to F n. It is denoted TGLnpFq. By Proposition 4.2.21 it is indeed a submonoid

of HompF n, F nq.

Theorem 4.3.7 Let F be a tropical semifield and n P N. The following statements aboutTGLnpFq hold.

(i) There is a dually residuated surjective homomorphism from TGLnpFq to the subsetMnpFq of matrices with finite tropical determinant. The dual residual maps thematrix A to the linking system induced by the weighted bipartite graph on prns, rnsqwith edge weights ti, ju ÞÑ Ai j .

(ii) Let K be an algebraically closed field equipped with a surjective valuation ω :

K Ñ F. For each λ P TGLnpFq there exists a matrix A P MnpKq such that λ isrealized by A, i.e. λpX , Y q � ωpdet ArX , Y sq. Equivalently, the actual morphismpidrns�λqpF

nq is an element of ωpGrnpK2nqq.

(iii) Every element λ P TGLnpFq induces a map Matprnsq Ñ Matprnsq that preservesmatroid rank and the quotient relation. Moreover, λ is uniquely determined by thismap. In particular, λ determines a rank preserving map between tropical linearsubspaces of F n.

Proof. We prove the claim in order stated.

(i) The map π : TGLnpFq Ñ MnpFq defined by λ ÞÑ pλptiu, t juqi j is well-defined.

To show that its dual residual is as stated, it is sufficient to show that for any linking

system κ with πpκq � πpλq and all X , Y � rns of equal cardinality κpX , Y q ¥ λpX , Y q.By an iterated application of the exchange condition for linking systems, with X2 �H

and Y2 �H the inequality,

κpX , Y q ¥äiPXκptiu, tσpiquq �

äiPXλptiu, tσpiquq ¥ λpX , Y q,

holds for some σ : X Ñ Y . This proves the statement.

(ii) This is an immediate consequence of Proposition 4.2.15 and the fact that the

restriction of a valuated matroid realizable over some valued field K is again realizable


over K .

(iii) The first statement is a specialization of Proposition 4.3.5. For the second

statement, let Γ be a directed weighted acyclic network in Γ that represents the mor-

phism λ. There is a vertex disjoint linking between the n source vertices and the nterminal vertices, that restricts to a matching of any subset of the source vertices to a

subset of equal size of the terminal vertices. Hence, rkλpµq � rkµ, for any matroid µ

on the source vertices. l

Example. This example serves to explain Figure 4.2 in detail. For convenience, denote

the domain set by R and the range set by S, say R� t1, 2u and S � t3,4u. A morphism

F2 Ñ F2 is a linking system on pR, Sq that assigns a finite value to the pair pR, Sq.Identifying such linking systems with their graphs (or representation matroids) yield

that TGL2pFq is the subset of Dr2pFR � FSq consisting of tropical linear spaces whose

image under the projection FR�FS Ñ FR is FR and whose image under the projection

FR � FS Ñ FS is FS . The condition that such a tropical linear space needs to be the

projection of an iterated linear extension is superfluous, since by Corollary 4.2.14,

any elementary extension is linear. In other words, there is a single directed network

Γ that yields all morphisms if we very the arc weights in F . It is depicted in Figure

4.12.

1 2

3

4

α1 α2

β3 β1 β2

Figure 4.12: Directed network inducing the morphisms in TGL2.

For example, to obtain the morphisms in the cone labeled 1 in Figure 4.2 choose

α1 � a, α2 � c, β1 � b, β2 � d and β3 � 8. To obtain the morphisms in the cone

labeled 3 one chooses α1 � a, α2 � c, β1 �8, β2 � e� a and β3 � b� a � d � c.

Any such choice of arc weights gives a linking system λΓrR, Ss. The matrix depicted

below the combinatorial type of the cone is,

�� λpt1u, t3uq λpt1u, t4uq 8

λpt2u, t3uq λpt2u, t4uq 8

8 8 λpt1,2u, t3,4uq

��


and hence TGL2 is a four-dimensional complex. Dividing out the actions of multiplica-

tion of the columns with a constant and the multiplication of the rows with a constant

gives the two dimensional complex depicted in Figure 4.2.

Complete intersections, transversal matroids and morphisms

An n�d dimensional tropical linear subspaceL of F n is called a complete intersectionif there exist d linear forms α1, . . . ,αd on F n such that L is the intersection of the

hypersurfaces αKi . The aim is to characterize the tropical linear subspaces that are

complete intersections, and use these results to study generic elements of TGLnpFq.

An essential ingredient in the characterization is Theorem 5.3. of First steps intropical geometry, which we will rephrase for convenience. Fix integers d ¤ n and let

A be a d � n matrix over F . Such a matrix determines a matroid ξA of rank d on rns,by the following expression,

ξApJq � detpAJq, J � rns and |J | � d.

Here AJ denotes the d�d submatrix of A consisting of the columns indexed by J . The

matroids that arise in such a way are the valuated analogues of transversal matroids,

but one can also think of them as matroids realizable over F .

Interpret the rows of A as linear forms α1, . . . ,αd on F n. Their intersection is a

tropical prevariety that contains the stable intersection of the hypersurfaces αKi . In

particular, it contains a tropical linear space of dimension n� d.

Lemma 4.3.8 Write LA for the linear space associated to ξA. The stable intersection ofthe hypersurfaces αKi equals LK

A .

Proof. We do the proof on the level of matroids. Let µi be the valuated matroid αi , i.e.

µip jq � αipe jq. By definition ξA is the matroid union of the matroids µi . This extends

the fact that a transversal matroid is a union of rank 1 matroids. The statement follows

by the facts that matroid intersection is dual to matroid union and that the linear

space of the intersection of matroids is the stable intersection of the associated linear

spaces. For the last statement see Tropical linear spaces. l

Now Theorem 5.3 establishes a necessary and sufficient condition for when LKA

equals the intersection of hyperplanes αKi .

Theorem 4.3.9 (Theorem 5.3) The tropical linear space LKA is contained in the inter-

section of hypersurfaces αKi . They are equal if and only if AJ is tropically non-singularfor all J � rns of size d.


Proof. The first statement is an immediate consequence of the previous lemma. By

another method, it is easy to see that αi is contained inLA, since it is the image of the

point ei under the linking system coming from the bipartite graph on prds, rnsq with

weight function wpi, jq � Ai j .

Proof of the second statement is much more complicated and we refer to reader to

First steps in tropical geometry. It would be nice to have a purely combinatorial proof

though. l

We are now ready to prove our main claim.

Theorem 4.3.10 An n� d dimensional tropical linear subspace L of F n is a completeintersection if and only if its dual is of the form LA, where A is a d � n matrix over Fsatisfying the property that AJ is tropically nonsingular for every J � rns of size d.

Proof. Suppose thatL �LKA . By Theorem 4.3.9 the intersection of hypersurfaces αKi

equals LKA , where α1, . . . ,αd are the linear forms coming from the rows of A. Thus,

L is the complete intersection of the hypersurfaces αKi .

Suppose that L is the complete intersection of hypersurfaces αKi , i � 1, . . . , d. Let

A be the d�n matrix whose rows are the αi and consider the tropical linear spaceLA.

Since αi PLA it follows that L �LKA . Now it is sufficient to show that dimLA � d.

The dimension of LA equals the rank of the underlying matroid MA of ξA. But MA

is the transversal matroid of the set system S � pS1, . . . , Sdq, where j P Si whenever

the j-th coefficient of αi is finite. We need to show that S has a transversal. Suppose

that it does not. By Hall’s Marriage theorem (cf. Theorem 12.2.1 of [50]) there exists

an X � rds satisfying |X | ¡ |SX |, where

SX �¤iPX

Si .

The intersection of all αKi with i P X equals its full preimage under the projection

onto the coordinates SX , since there are no conditions on the coordinates outside of

SX . Hence it is a prevariety of dimension larger than n� |X |, contradicting the fact

that L is a complete intersection.

Thus, L � LKA and, by Theorem 4.3.9, the matrix A satisfies the required condi-

tion. l

Remark. The theorem above generalizes Theorem 6.3 of The tropical Grassmannianby providing both a converse and extending to arbitrary dimension. Note that in the

rank 2 case the projectivization of a linear space of the formLA is almost a caterpillar

tree, save for the fact that different legs might be attached to the same point of the

body. The condition that A is tropically non-singular precludes this. This is most easily

seen by considering ξA as the matroid obtained by extending the free matroid on


two elements by linear functions represented by the columns of A, say βi : F2 Ñ F ,

i � 1, . . . , n and deleting the two original coordinates. The tropical non-singularity

condition states that for two such functions βi1dβ j2 � βi2dβi1. This is equivalent to

the condition that βKi � βKj .

The application to the HompF n, F nq is via Proposition 4.2.22.

Theorem 4.3.11 Let λ P HompF n, F nq be a generic morphism in the sense that thematrix pλprnsztiu, rnszt juqi j satisfies the genericity condition of Theorem 4.3.10. Thenthere exists m¥ n and a tropical linear space L � F m such that,

(i) L is an iterated linear extension of F n,(ii) the image of L under the canonical projection F m Ñ F n equals pidn �λqpF

nq,(iii) the orthogonal dual of L has ξA as associated matroid, with A a matrix over F.

Proof. By definition of a morphism there exists an iterated linear extension L of F n.

This iterated linear extension is of the form pidB � λΓrB, RzBsqpF nq for some linking

system λΓ induced by the weighted directed network Γ on a set of nodes R, where Bis the set of n source nodes of Γ. Thus, L � λΓpF

Bq and a matroid µ associated to Lis Y ÞÑ λΓpB, Y q. The dual satisfies,

µKpZq � µpRzZq � λΓpB, RzZq � λΓKpRzB, Zq � λ∆pRzB, Zq,

for some weighted bipartite graph ∆ on pR, Rq. However, since the only nodes in a

basis on the domain nodes of ∆ are RzB it holds that λ∆pRzB, Zq � λ∆1pRzB, Zq,with ∆1 the induced subgraph of ∆ on pRzB, Rq. Thus, µK equals µA, where A is the

pRzBq � R-matrix over F with pi, jq-th entry λ∆ptiu, t juq � λΓpRztiu, Rzt juq. l

Chapter 5

The gossip monoid

Suppose somebody wishes to travel between three locations: the center of Eindhoven

(E), the center of Amsterdam (A) and the boundary of Amsterdam (B), and has two

types of transportation at its disposal: a bicycle, and a car. The time (in minutes) it

takes to travel between the different sites is given in the tables below.

bicycle E B A

E 0 630 640

B 630 0 20

A 640 20 0

car E B A

E 0 90 140

B 90 0 60

A 140 60 0

The car has a bike rack. This is convenient when travelling from E to A, since one

can take the car (and bicycle) to B and proceed on bicycle to A. The total travelling

time is 110 minutes. However, it doesn’t help when travelling from A to E. In that case

it is fastest to leave the bike and go by car, for a total travelling time of 140 minutes.

The time it takes to travel between A, B and E in this setup is given in the next table.

car and bicycle E B A

E 0 90 110

B 90 0 20

A 140 20 0

This is not a metric, since the travelling times are not symmetric and do not satisfy

the triangle inequality.

In this chapter we study the monoid of finite tropical products of distance matri-

ces. In this setting the situation above is represented by taking the product of two

distance matrices, one corresponding to the bike metric, the other to the car metric.The example shows that the tropical product of distance matrices is not necessarily a

127

128 CHAPTER 5. THE GOSSIP MONOID

distance matrix.

Section 5.1 gives the basic definitions, and establishes some easy results. Section

5.2 gives the relation with the gossip problem, discussed for example in [3]. It ex-

plains the name of this chapter. Section 5.3 deals with a possible way of giving a

polyhedral fan structure to the monoid. And finally, Section 5.4 explicitly constructs

the monoid generated by the distance matrices of size 3� 3 and that of size 4� 4.

5.1 Foundations

Let F be a tropical semifield and n a natural number. In this chapter it is more nat-

ural to write the semifield operations in the classical manner, and hence we have an

explicit inclusion F � R Y t8u such that x d y � x � y and x ` y � mintx , yu.Consider the semiring MnpFq of square matrices of size n with entries in F under the

operations of tropical matrix multiplication d and tropical matrix addition `, defined

by the expressions,

�X d Y

�pi, jq � mintX pi, kq � Y pk, jq | k � 1, . . . , nu, (5.1)�

X ` Y�pi, jq � mintX pi, jq, Y pi, jqu (5.2)

where X , Y P MnpFq. Moreover, when Fzt8u is a ring under the ordinary operations

restricted from R, the Fzt8u-module structure of MnpFzt8uq � MnpRq extends natu-

rally to MnpFq. We denote the extended scalar multiplication by λ�X and the extended

matrix addition by X�Y , for λ P F . We refer to them as the ordinary or classical matrix

operations.

Remark. It is convenient to interpret the entry at position pi, jq of an element of thesemiring MnpFq as the cost of moving from site i to site j. In this interpretation the

product X d Y of two such elements X and Y has as entries the minimal cost of

moving from one state to another in precisely two steps, where the costs of the first

step are determined by X , and the cost of the second step are determined by Y . The

interpretation of an m-fold (finite) product is similar. The entries of the sum X ` Yare the minimal costs of moving from site to site where one can choose to calculate

costs either according to X , or to Y .

A symmetric matrix X P MnpFq is said to be a distance matrix if its entries are

non-negative, satisfy the triangle inequality,

X pi, kq ¤ X pi, jq � X p j, kq, i, j, k P rns (5.3)

and are equal to 0 on the diagonal. Note that the non-negativity criterion is a con-

sequence of the other two by taking i � j � k in (5.3). Denote the collection of

5.1. FOUNDATIONS 129

distance matrices by DnpFq. Since the number of inequalities in (5.3) is finite, DnpFqis a polyhedral complex. It is also a cone in MnpFq in the sense that (i) it is closed

under ordinary scalar multiplication by non-negative elements of F , (ii) it is closed

under ordinary matrix addition. However, by the following example, DnpFq is neither

closed under tropical matrix addition, nor under tropical matrix multiplication.

Example. Let X , Y P DnpBq, where B is the trivial semifield t0,8u, be given as,

X �

�� 0 0 8

0 0 8

8 8 0

�� , Y �

�� 0 8 8

8 0 0

8 0 0

��

Then Z � X ` Y is not a distance matrix, since 8 � Zp1,3q ¦ Zp1,2q � Zp2, 3q � 0.

Moreover, X d Y is the non-symmetric matrix,

X d Y �

�� 0 0 0

0 0 0

8 0 0

��

Remark. Equip MnpFq with the order relation ¤ that is defined by X ¤ Y when

X i j ¤ Yi j , for all i, j P rns. We write I for the multiplicative neutral element, i.e. the

tropical identity matrix, with entries

Ii j �

#0 if i � j,8 if i � j

,

and Jλ for the matrix all of whose entries are λ, for some λ P F . The set of distance

matrices is equivalently characterized as the subset of the symmetric matrices in the

closed interval rJ0, Is that are idempotent under tropical matrix multiplication.

The submonoid of pMnpFq,dq generated by DnpFq is denoted SnpFq and referred

to as the gossip monoid or the gossip monoid on n points, for reasons to be explained

later. By the previous example, SnpFq is in general strictly bigger than DnpFq. The next

example shows that there exist matrices all of whose entries are non-negative that are

not in SnpFq.

Example. The following illustrates that not every element of MnpFq is in the monoid

SnpFq generated by the elements of DnpFq. The simplest such example occurs for

n� 2. The matrix,

X �

�0 8

0 0

�


is not a product of distance matrices. Suppose that it were. Write X � X1 d . . .d Xm.

Since n � 2 there is some matrix X j for which the p2,1q-st entry equals 0. But then

the p1, 2q-st entry of X must be equal to zero, by the fact that both right and left

multiplication by matrices with diagonal equal to 0 are decreasing maps.

The gossip monoid is generated by a finite number of one parameter submonoids,

which we define now. Let i, j P rns with i � j and a P F . The matrix Ei jpaq whose

pk, lq-th entry is, $'&'%

0 if k � l,8 if k � l and tk, lu � ti, jua if tk, lu � ti, ju

In other words, it is the matrix that differs from the tropical identity matrix only at

entries pi, jq and p j, iq, where it is a. Since the triangle inequalities (5.3) are trivially

satisfied, Ei jpaq is a distance matrix. A matrix of the form Ei jpaq is referred to as

an elementary distance matrix. The next statement shows that they indeed suffice to

generate SnpFq.

Lemma 5.1.1 The gossip monoid SnpFq is generated by the elementary distance matri-ces.

Proof. It is sufficient to show that any element of DnpFq can be written as a product

of elementary distance matrices. Let X P DnpFq. We claim that,

X �äi j

Ei jpX i jq, (5.4)

where the product is taken in any order. Let k, l P rns and consider the entry at position

pk, lq of the right-hand side of the above expression. By elementary matrix algebra it

equals,nà

i1�1. . .

nàir�1

�pEI1

qki1 � pEI2qi1 i2 . . .� pEIr�1

qir�1 ir� pEIr

qir l

, (5.5)

where I1, . . . , Ir are the size 2 subsets of rns in the order of (5.4). Consider a finite

summand of the above expression corresponding to the tuple I1, . . . , Ir . Then k � i1,

or I1 � tk, i1u, and i j�1 � i j or I j � ti j�1, i ju, and ir � l, or Ir � tir , lu. Let pk �j0, j1, . . . , jm � lq be a maximal, non-staggering subsequence of pk, i1, . . . , ir , lq, i.e.

js � js�1 for all s, and maximal as such. Then the summand under consideration

equals,

X j0, j1 � X j1, j2 � . . .� X jm�1, jm ,

which is at least Xkl by the triangle inequality. Since Xkl also appears in (5.5) the

statement follows. l

5.1. FOUNDATIONS 131

Let A P SnpFq. The length `pAq of A is the minimal length of a product of elementary

distance matrices equal to A, or more precisely,

`pAq �mintm | there exist elementary distance matrices X1, . . . , Xm

such that A� X1 d . . .d Xmu (5.6)

There is an easy bound on the length of an element of SnpFq.

Proposition 5.1.2 The length of an element of SnpFq is at most n3 � n2.

Proof. Let X be an arbitrary finite product of elementary distance matrices, i.e.,

X �mä

k�1

EIkpakq,

for sequences I1, . . . , Im of pairs and a1, . . . , am P F . The entry at position pi, jq of X is

ak1� . . .� aks

, where pIk1, . . . , Iks

q is a path from i to j in the complete graph on rns.Since revisiting a site is never cheaper, without loss of generality s ¤ n. The bound

follows since X has at most n2 � n non-zero entries pi, jq. l

To make use of the techniques available for representing a distance matrix by a

graph it is useful to have the following construction. To a matrix A we associate the

matrix A� of minimal costs of moving from one site to another in any number of steps.

Algebraically, this translates to,

A� � I ` A` A2 ` . . .� I ` A` A2 ` . . .` An, (5.7)

where exponentiation is with respect to d. The latter equality holds since it is never

cheaper to revisit a site, by the positivity of the entries of A. Incidentally, the opera-

tor A ÞÑ A� makes MnpPq into a structure known as a Kleene algebra. The relevant

properties of the operator are given in the next lemma.

Lemma 5.1.3 Let A P MnpFq. The matrix A� is idempotent with diagonal entries 0.If A has nonnegative entries and 0 on the diagonal then A� � An. If A P SnpPq thenA� P DnpPq.

Proof. Let A P MnpFq. The idempotency follows by expanding the product A� d A�,

making use of (5.7). Let X P MnpFq have non-negative off-diagonal entries and di-

agonal entries 0 and let Y P MnpFq have non-negative entries. Consider the entry at

position pk, l) of the product X d Y . It equals,

nài�1

�Xki � Yilq � Ykl `

� nài�1i�k

Xki � Yil

.


Thus, X d Y ¤ Y . It follows that An ¤ Am, with m¤ n and hence A� � An, when A has

diagonal entries 0.

By the algebraic characterization of distance matrices given earlier it is sufficient to

show that A� is symmetric. By Lemma 5.1.1 there exist elementary distance matrices

X I1, . . . , X Ir

such that,

A�rä

j�1X I j

.

The entry at position pk, lq of A� � An equals the minimum of all AJ1� . . . � AJm

,

where the Ji P tI1, . . . , Iru and pJ1, . . . , Jmq form an edge path between k and l in the

complete (loopless) graph on rns. Then AJm� . . .�AJ1

is at least A�lk, since pJm, . . . , J1q

is a path from l to k. Thus, A�lk ¤ A�kl and the statement follows by symmetry. l

5.2 Relation to the gossip problem

We refer to SnpFq as the gossip monoid, as the case F � B is strongly related to the

gossip problem. This mathematical problem appears to be first published in [3] and is

stated there as follows.

There are n ladies, and each of them knows some item of gossip not

known to the others. They communicate by telephone, and whenever one

lady calls another, they tell each other all that they know at that time.

How many calls are required before each gossip knows everything?

In our setting, it is the problem of determining the length of the all-zero matrix

J0. A scheme of calls solving the gossip problem translates to an expression for J0

as a product of a minimal number of elementary distance matrices Ei j , where in this

section Ei j is shorthand for Ei jp0q. Each such elementary distance matrix represents a

phone call between person i and j. The following example serves to clarify the link.

Example. Let n � 4 and consider the tropical product E12 d E23 d E34. The product

represents a sequence of three calls starting with a call between 1 and 2, then a call

between 2 and 3, and ending with a call between 3 and 4. The pieces of gossip 1

knows after this sequence are his own, and that of 2, the pieces of gossip 2 knows

after the sequence is that of 1, that of 2 and that of 3, while both 3 and 4 know all

the gossip. This is reflected in the columns of the product E12 d E23 d E34, i.e.

E12 d E23 d E34 �

��

0 0 0 0

0 0 0 0

8 0 0 0

8 8 0 0

��

5.2. RELATION TO THE GOSSIP PROBLEM 133

In the article [3] a simple solution to the gossip problem is given. The authors

show that any product of less than 2n� 4 fundamental generators is strictly bigger

than J0, for n ¥ 4. Their solution depends on four chief gossips, an initial collector

i and a final spreader j, both of whom are chief gossips. Assume without loss of

generality that the chief gossips have labels 1, 2, 3 and 4. Then,

J0 �� nä

k�5

Eik

d XC d

� näk�5

E jk

where XC represents a calling scheme among the chief gossips to distribute the in-

formation among themselves. It has length 4 and can be taken equal to the product

E12 d E34 d E13 d E24. The length of J0 for n� 1,2, 3 is 0,1, 3.

There are some questions related to the gossip problem that are very natural from

the point of view of the gossip monoid, that to our knowledge have not even been

asked, let alone answered, in published sources.

(i) What is the maximal length of an element of SnpBq?(ii) What is the size of SnpBq?

(iii) Is there an efficient way to determine whether an n � n-matrix over B is an

element of SnpBq?Table 5.1 presents some computational results on questions (i) and (ii) obtained by

Jochem Berndsen in [6]. Notice that for n � 6,7, 8 the all-zero matrix is not an

element of maximal length in SnpBq (in those cases `pJ0q � 8, 12,14). Note that there

is rather large gap between the values in the table and the simple upper bound of

n3 � n of Proposition 5.1.2.

n |SnpBq| max. length1 1 02 2 13 11 34 189 45 9152 66 1,092,473 107 293,656,554 138 166,244,338,221 16

Table 5.1: Sizes and maximal lengths of SnpBq, for n� 1, . . . , 8.

Remark. The monoid SnpR8q can be seen as modelling a related real world situation

of sharing n gossips between n gossipers. The setup is the same is for the B case, with

the added complication that gossipers now know a certain fraction of each gossip.


Moreover, phone calls are over an imperfect channel, in the sense that they only

transmit a fixed fraction of the information between both participants. More precise,

an elementary distance matrix Ei jpaq corresponds to a phone call between i and j in

which a fraction e�a of the information is interchanged between gossip i and j.

5.3 Polyhedral structure of the gossip monoid

In this section we restrict our attention to the case F � R8. The following theorem

exploits the fact that SnpR8q is contained in a tropicalization of the orthogonal group.

Theorem 5.3.1 The elements SnpR8q X Rn2are contained in the underlying set of a

polyhedral fan of dimension pn2q.

Proof. Let K � Ctttuu be the field of Puiseux series over C in t with order valuation

ord : K ÑQ8. We prove that the matrices in SnpR8qwith entries in Q8 are contained

the tropicalization of OnpKq. The statement then follows by the Bieri-Groves theorem

(Proposition 1.4.9) and Proposition 1.2.4. The polyhedral fan in the statement is com-

mon refinement of the unions of the polyhedral fan structures on the tropical varieties

of the irreducible components of OnpKq.

Let x P K and i, j P rns with i � j. Define the matrix Gi jpxq by,

Gi jpxqpk, lq �

$''''''&''''''%

cospxq if k � l � i, or k � l � j,� sinpxq if k � i and l � j,sinpxq if k � j and l � i,0 if k � l and tk, lu � ti, ju,1 if k � l and k, l R ti, ju.

Then Gi jpxq P OnpKq and ordpGi jpxqq � Ei jpordpxqq.

Let I1 � pi1, j1q, . . . , Ir � pir , jrq be a sequence of pairs and a1, . . . , ar P K . Consider

the product,

X �rä

k�1

EIkpakq.

Since the residue field C of K is infinite, there exist t1, . . . , t r P K such that ordptkq �

ak and

ord� r¹

k�1

GIkptkq��

räk�1

ord�GIkptkq�� X .

Thus, X P Trop pOnq and SnpR8q is contained in the topological closure of Trop pOnq.

l

5.3. POLYHEDRAL STRUCTURE OF THE GOSSIP MONOID 135

The theorem does not give a natural polyhedral structure on SnpR8q. We attempt

to give SnpR8q a natural polyhedral structure in the next section.

5.3.1 Graphs with detours

The next part is concerned with the problem of realizing an element of MnpR8q by a

graph with a finite number of prescribed paths between the nodes. When restricted to

elements of DnpR8q this realization problem is that of embedding a metric on a finite

number of points in a weighted graph (cf. [34]), which we explain first.

Let Γ� pV, Eq be an undirected graph and w : E Ñ F a function assigning nonneg-

ative weights to the edges. We do not assume Γ to be simple. The weight of a path in

pΓ, wq is the sum of the weights of the individual edges in the path. It is thought of as

the cost of traversing the path. A map ` : rns Ñ V is called a labeling, or rns-labeling,

if we need to be precise, and the pair pΓ,`q is referred to as a labeled graph, or an

rns-labeled graph.

A weighted rns-labeled graph gives rise to a matrix ApΓ, w,`q in DnpFqwhose entry

at position pi, jq is the minimal weight of a path between `piq and `p jq. We say that

the weighted labeled graph pΓ, w,`q realizes a matrix X if X � ApΓ, w,`q. Moreover,

such a realization is said to be optimal if the total sum of the edge weights of pΓ, wq is

minimal among all realizations. Optimal realizations always exist, as proven in [34].To exclude superfluous edges we require that an optimal realization of X is minimalin the sense that the result of a contraction or deletion of any edge does not realize

X .

The terminology extends to labeled graphs without weights. Such a graph is said

to realize a certain matrix when there exist a weight function on its edges that realizes

the matrix in the previously defined sense. The qualifiers optimal and minimal extend

in the same manner.

In what follows we restrict our attention to F � R8. The first step in describing

the cones of the gossip monoid is to find a minimal set of labeled graphs that realize

the elements of DnpR8q optimally (and thus minimally, by definition). We do so in

the next example. See also [22].

We remind the reader that J0 stands for the matrix of the appropriate size with all

entries 0.

Examples. We give optimal realizations of the elements of DnpR8q, for n � 2,3, 4.

For the cases n� 5,6 see [41] and [63].

(i) Optimal realizations of the elements of D2pR8q are easily determined. An ele-

ment of D2pR8qztJ0u is minimally realized by the graph on two vertices having

one edge. The choice of labeling is inconsequential as long as it is injective. The

matrix J0 is optimally realized by the graph on one vertex.


1 323 32 21 1

1, 2 3 1, 3 2 2, 3 1

1, 2, 3

1 3

2

Figure 5.1: The Hasse diagram of the poset of minimal graphs that optimally realizethe elements of D3pR8q.

(ii) We consider the case n� 3. Any matrix in D3pR8q is realized by the top labeled

graph of the poset depicted in Figure 5.1, although only the matrices in the

interior of the cone are optimally realized by it. However, any matrix of D3pR8qis minimally realized by some graph in the poset. The graph depends on the

facet of D3pR8q the matrix is contained in. In effect, the order complex of the

poset of Figure 5.1 is isomorphic to the polyhedral cone D3pR8q.

1 2

3 4

1 2

34

1

2

3

4

Figure 5.2: The three distinct labeled graphs that realize all matrices in D4pR8q Theparallel sides of the middle rectangle have equal weight.

(iii) The case n � 4 is similar to that of n � 3 in the sense that there exists a single

graph Γ such that for any element X of D4pR8q there exists a labeling ` of Γsuch that X is realized by pΓ,`q. However, three distinct labelings are required,

unlike in the case n� 3. The labeled graphs are depicted in Figure 5.2.

We wish to extend the representation of DnpR8q in terms of realizing labeled

graphs to the entire monoid SnpR8q. For this we need an extension of the concept

5.3. POLYHEDRAL STRUCTURE OF THE GOSSIP MONOID 137

of a labeled weighted graph. Let i and j be distinct elements of rns. A detour from ito j in an rns-labeled weighted graph is simply a path p starting at `piq and ending

at `p jq that has larger total weight than the path of minimal weight between `piqand `p jq. Note that a path is allowed to traverse the same edge more than once. The

data specifying the detour is the triple pi, j, pq. A labeled weighted graph with detoursis a tuple consisting of a labeled weighted graph and a finite set of detours between

distinct pairs of vertices.

Let pΓ, w,`,Dq be an rns-labeled weighted graph with set of detours D. It gives rise

to a matrix ApΓ, w,`,Dq whose entry at position pi, jq equals the weight of the detour

from i to j, if there is any, or the weight of a path of minimal weight between i and j,if there is no detour between i and j in D. In particular, ApΓ, w,`,Dq need not be sym-

metric, while its diagonal entries must be 0. Again, if X P MnpFq and X � ApΓ, w,`,Dq,then pΓ, w,`,Dq is said to realize X and similar terminological considerations as for

the detourless case apply. In particular, a matrix X is said to be realized by a triple

pΓ,`,Dq if there exists an assignment of edge weights w to Γ such that pΓ, w,`,Dqis a weighted labeled graph with detours and X � ApΓ, w,`,Dq. By slight abuse of

terminology we refer to the triple pΓ,`,Dq as a labeled graph with detours.

Examples. We give three examples of labeled (weighted) graphs with detours.

1 2va b

(a) Linear graph with a sin-gle detour from 1 to 2.

1

2

3 4

(b) Graph with a single detour from 1to 4.

1, 2, 3

3 → 1

(c) A r3s-labeled graph onone vertex and one edge(a loop) with a single de-tour from 3 to 1.

Figure 5.3: Examples of labeled weighted graphs with detours.

(i) The first example is very simple. Consider the linear graph pΓ,`q on three ver-

tices one of whose end vertices is labeled 1 and the other 2. For ease of notation

we label the middle vertex v and define p to be path pt1, vu, t1, vu, t1, vu, tv, 2uq

from 1 to 2. Set the weight wpt1, vuq � a and wptv, 2uq � b. It is represented


in Figure 5.3(a). The matrix it gives rise to is,

ApΓ, w,`, tpuq �

�0 3a� b

a� b 0

�

Unless a � 0, or a � 8 this matrix is not in S2pR8q � D2pR8q. The matrices

that can be realized by pΓ,`, tpuq are precisely those in the cone tX P M2pR8q |X11 � X22 � 0, X12 ¥ X21, and X21 8uY I .

(ii) An example of a labeled graph with detours is displayed in 5.3(b). In Section

5.4.2 we will see that it realizes an element of S4pR8q. Note that the matrix Xrealized by this graph satisfies X41 � X13 � X34 ¤ X14 ¤ X12 � X24.

(iii) The third example is displayed in Figure 5.3(c). The graph Γ has a single point,

say v, and a single edge e connecting v to itself. The data specifying the detour

is the triple p3, 1, eq, in which 3 is the start and 1 is the end of the detour.

This detour graph appears in the classification of the cones of S3pR8q. A matrix

realized by this detour graph has 0 at all positions except at position p3, 1q,

where it may have any value in R8.

The following easy statement allows one to realize the transpose of a matrix real-

ized by a labeled weighted graph with detours.

Lemma 5.3.2 Let pΓ, w,`,Dq be a weighted rns-labeled graph with detours. Write D 1

for the set of detours,

D 1 � tp j, i, pq | i, j P rns and pi, j, pq P Du.

Then ApΓ, w,`,D 1q is the transpose of ApΓ, w,`,Dq.

Proof. Set X � ApΓ, w,`,Dq and Y � ApΓ, w,`,Dq. If there are no detours from i to

j, or from j to i then X i j � X ji � Yji � Yi j . If there is a detour from i to j in D, then

there is a detour from j to i in D 1, with the same path. Thus, X i j � Yji . This proves

the statement. l

The matrices realized by a labeled (unweighted) graph with detours G � pΓ,`,Dqform a polyhedral cone that is a linear image of pR�8q

E , where E is the set of edges of

Γ. This cone is referred to as the cone of G and denoted C pGq � MnpR8q. We aim to

find a finite set of labeled graphs with detours tG1, . . . , Gku such that,

SnpR8q �k¤

i�1

C pGiq. (5.8)

As the examples in the next section show, for n � 2,3, 4 the Gi can be chosen such

that C pGiq are the maximal cones of a fan structure on SnpR8q. We do not know if

5.4. CASES 139

such a choice is possible in general.

5.4 Cases

In this section we determine the gossip monoids on 3 and 4 points explicitly. Up to

symmetry there are only 2 distinct maximal cones in the fan S3pR8q, while there seem

to be 16 in S4pR8q.

5.4.1 n¤ 3

Since Ei jpaq d Ei jpbq � Ei jpa � bq and S2pR8q � D2pR8q it is clear that S2pR8q is

isomorphic to the monoid of positive tropical real numbers under addition (tropical

multiplication). The first non-trivial case is that of S3pR8q, the structure of which we

determine in this section.

Since S3pR8q is a pointed fan no information is lost by intersecting S3pR8q with

a sphere centered around the point of the fan (which is J0). We denote the resulting

spherical polyhedral complex by S13pR8q. It is depicted in Figure 5.4. The maximal

cones are those consisting of the matrices that are realized by the labeling detour

graphs.

1

23

1 3 2

3 2 1

3 2 1

1 3 2

1 3 2

1 3 2

Figure 5.4: Representation of the spherical complex S13pR8q. The labeled graphs withdetours corresponding to the maximal cells are indicated. The middle triangle repre-sents the cone of distance matrices.

The computations to show that Figure 5.4 is a represention of S13pR8q are elemen-

tary and can be done by hand. To prove that the matrices realized by the graphs with


a single detour in Figure 5.4 are indeed in S3pR8q it suffices to check that,

Api j ka b

c

q � E jkpbq d Api j k

a c− aq � Ap

i j k

c− b bq d Ei jpaq, (5.9)

for any valid choice of a, b and c. Notice that,

Api j ka b

c

q d Ei jpdq, and E jkpdq d Api j ka b

c

q

are contained in the complex of Figure 5.4 for all choices of a, b, c and d by (5.9).

Thus, to see that these cones are all it is sufficient to check that,

Ei jpdq d Api j ka b

c

q, �

$''''''''&''''''''%

Api j ka b

c

q, c� b ¤ d,

Api j ka b

b + d

q, a ¤ d ¤ c� b,

Api j kb

a + b

dq, 0¤ d ¤ a

,

Eikpdq d Api j ka b

c

q �

$''''''''''''''''&''''''''''''''''%

Api j ka b

c

q, c ¤ d,

Api j kb

d

aq, a� b ¤ d ¤ c,

Api j

k

a+b−d2

b+d−a2

a+d−b2

q, b� a ¤ d ¤ a� b,

Apij k

b

daq, 0¤ d ¤ b� a.

(when a ¤ b),

5.4. CASES 141

Eikpdq d Api j ka b

c

q �

$'''''''''''''''&'''''''''''''''%

Api j ka b

c

q, c ¤ d,

Api j kb

d

aq, a� b ¤ d ¤ c,

Api j

k

a+b−d2

b+d−a2

a+d−b2

q, a� b ¤ d ¤ a� b,

Apij k bd

a

q, 0¤ d ¤ a� b.

(when b ¤ a).

That the products,

Api j ka b

c

q d Eikpdq, and Api j ka b

c

q d E jkpdq

are also contained in one of the cones of Figure 5.4 follows by transposing the product

matrices and applying Lemma 5.3.2.

5.4.2 n� 4

The case n � 4 is sufficiently large to make the computations intractable by hand.

Instead we used Mathematica to compute a fan structure on SnpR8q. Figure 5.5 gives

the realizing graphs with detours of all the cones of S4pR8q, up to transposition and

the action of Σ4, the symmetric group on t1, 2,3, 4u. The surplus length of a detour

from i to j is defined as the difference between the length of the detour and the

minimal distance between i and j in the graph. Two detours between i to j and k to

l have the same color if their surplus lengths are equal.

These graphs were obtained by generating all 66 possible piecewise linear affine

maps R68Ñ M4pR8q of the form

px1, . . . , x6q Ñ EI1px1q d EI2

px2q d . . .d EI6px6q,

up to Σ4 and transposition, and determining the �-maximal cones in their images. It

then suffices to check that the collection of cones obtained in such manner is closed

under left multiplication by an elementary distance matrix. Connectivity in codimen-

sion 1 is proved by Figure 5.6. It shows that any maximal cone is connected in codi-

mension 1 to at least on cone of distance matrices. Note that each labeling depicted

in Figure 5.2 corresponds to a different maximal cone. Since the distance matrices

are themselves connected in codimension one it follows that S4pR8q is connected in

codimension 1.

Most intersections in Figure 5.6 are of a special type. Recall that a cone determined


by a graph with detours is the image of a linear map pR�8qE Ñ MnpR8q, where E is

the set of edges of the graph. According to Figure 5.5 every cone C in SpR8q is the

image of such a map αC with domain pR�8q6. Moreover, it is simple to check that all

cones are six dimensional. In particular, the intersection of such a cone with a sphere

around the origin is a simplex. Most intersections in Figure 5.6 are of the image of the

restriction of αC to a five dimensional boundary hyperplane of pR�8q6, in other word,

they are obtained by setting one coordinate equal to 0. In terms of the graphs this is

expressed by contracting an edge. The only exception is the connection of C7 to C12.

Although these cones intersect in a five dimensional boundary cone, the boundary

cone is obtained from the parameterizations specified by the graphs with detours by

restricting them to a hyperplane of the form x i � x j .

This leads to the following theorem.

Theorem 5.4.1 The cones realized by the graphs of Figure 5.5 give a polyhedral fanstructure on S4pR8q. This polyhedral fan is pure of dimension 6 and connected in codi-mension 1. Its intersection with a sphere around the origin is a simplicial spherical com-plex. Moreover, every element of SnpR8q is the product of (at most) 6 elementary distancematrices.

We end this chapter with a conjecture. The results of this section give evidence for

the following statement.

Conjecture 5.4.1 There exists a polyhedral fan structure on SnpR8q such that this fanis pure of dimension pn

2q and connected in codimension 1. Every element of SnpR8q is the

product of (at most) pn2q elementary distance matrices.

5.4. CASES 143

C1 C2 C3

C4 C5 C6

C7 C8 C9

C10 C11 C12

Figure 5.5: The graphs realizing a polyhedral fan structure on S4pR8.


2

3 4

1 ∗

C2C10

1 2

34

∗

1

2

34

1

2

3 4∗

C9

1

234

∗

1

234

∗

4 3 2

1

C1 C2

12

3

4

1 2

3

4

1

23

4

∗∗

C8 C11

1

2

3

4

1

2

34

1

2

34

∗∗

C5 C4

1 2

34

12

34

∗

4

∗1

23

C12C11

1

2

34

1

2

343

1 4

2

∗ ∗

C4 C8

12

34

∗

1

3 4

2

1 2

3

4∗

C6 C3

1

2

3 4

C2

1 2

34∗

C4

1

2

34

∗

1 2

3

4

∗

C3

C7 C12

1

2

34

12

34

12

34

Figure 5.6: Intersection of cones realized by labeled graphs that are themselves re-alized by a contraction of the labeled graph (except in case C7-C12. The edge to becontracted is indicated by an asterix �. This shows that the cones in the grey boxesintersect in a cone of dimension 5. The intersection between C7 and C12 is obtainedby setting equal surplus lengths in the graphs representing C7 and C12.

Bibliography

[1] Marianne Akian, Stephane Gaubert, and Alexander Gutermann. Linear inde-

pendence over tropical semirings and beyond. Preprint, 2008.

[2] Federico Ardila and Caroline J. Klivans. The Bergman complex of a matroid and

phylogenetic trees. J. Combin. Theory Ser. B, 96(1):38–49, 2006.

[3] Brenda Baker and Robert Shostak. Gossips and telephones. Discrete Math.,2(3):191–193, 1972.

[4] George M. Bergman. The logarithmic limit-set of an algebraic variety. Trans.Amer. Math. Soc., 157:459–469, 1971.

[5] George M. Bergman. A weak Nullstellensatz for valuations. Proc. Amer. Math.Soc., 28:32–38, 1971.

[6] Jochem Berndsen. Three problems in algebraic combinatorics. Master’s thesis,

Eindhoven University of Technology, 2012.

[7] Robert Bieri and J. R. J. Groves. The geometry of the set of characters induced

by valuations. J. Reine Angew. Math., 347:168–195, 1984.

[8] T.S. Blyth and M.F. Janowitz. Residuation Theory. Pergamon Press, 1972.

[9] Alexandre V. Borovik, I. M. Gelfand, and Neil White. Coxeter matroids, volume

216 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2003.

[10] S. Bosch, U. Güntzer, and R. Remmert. Non-Archimedean analysis, volume 261

of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles ofMathematical Sciences]. Springer-Verlag, Berlin, 1984. A systematic approach to

rigid analytic geometry.

[11] Thomas Brylawski. Constructions. In Theory of matroids, volume 26 of Encyclo-pedia Math. Appl., pages 127–223. Cambridge Univ. Press, Cambridge, 1986.

145

146 BIBLIOGRAPHY

[12] Alan L. C. Cheung. Adjoints of a geometry. Canad. Math. Bull., 17(3):363–365;

correction, ibid. 17 (1974), no. 4, 623, 1974.

[13] Guy Cohen, Stéphane Gaubert, and Jean-Pierre Quadrat. Duality and separa-

tion theorems in idempotent semimodules. Linear Algebra Appl., 379:395–422,

2004. Tenth Conference of the International Linear Algebra Society.

[14] H.H. Crapo. Single element extensions of matroids. J. Res. Natl. Bureau Stan-dards B, pages 55–65, 1965.

[15] R.A. Cuninghame-Green and P. Butkovic. Bases in max-algebra. Linear Algebraand its Applications, 389:107–120, 2004.

[16] Raymond Cuninghame-Green. Minimax Algebra. Springer-Verlag, 1979.

[17] Vladimir I. Danilov, Alexander V. Karzanov, and Gleb A. Koshevoy. Planar flows

and quadratic relations over semirings. Journal of Algebraic Combinatorics,36(3):441–474, 2012.

[18] Mike Develin, Francisco Santos, and Bernd Sturmfels. On the rank of a tropical

matrix. In Combinatorial and computational geometry, volume 52 of Math. Sci.Res. Inst. Publ., pages 213–242. Cambridge Univ. Press, Cambridge, 2005.

[19] Mike Develin and Bernd Sturmfels. Tropical convexity. Documenta Mathematica,

9:1–27, 2004.

[20] Alicia Dickenstein, Eva Maria Feichtner, and Bernd Sturmfels. Tropical discrim-

inants. J. Am. Math. Soc., 20(4):1111–1133, 2007.

[21] Jan Draisma. A tropical approach to secant dimensions. J. Pure Appl. Algebra,

212(2):349–363, 2008.

[22] Andreas Dress, Katharina T. Huber, and Vincent Moulton. Hereditarily opti-

mal realizations: why are they relevant in phylogenetic analysis, and how does

one compute them? In Algebraic combinatorics and applications (Gößweinstein,1999), pages 110–117. Springer, Berlin, 2001.

[23] Andreas W. M. Dress and Walter Wenzel. Valuated matroids. Adv. Math.,93(2):214–250, 1992.

[24] Manfred Einsiedler, Mikhail Kapranov, and Douglas Lind. Non-Archimedean

amoebas and tropical varieties. J. Reine Angew. Math., 601:139–157, 2006.

[25] Georges François and Johannes Rau. The diagonal of tropical matroid varieties

and cycle intersections. Collectanea Mathematica, pages 1–26, 2012.

BIBLIOGRAPHY 147

[26] David Gale. Optimal assignments in an ordered set: An application of matroid

theory. J. Combinatorial Theory, 4:176–180, 1968.

[27] S. Gaubert and R. Katz. The Minkowski theorem for max-plus convex sets.

Linear Algebra and Applications, 421:356–369, 2007.

[28] Israel M. Gelfand, Mikhail M. Kapranov, and Andrei V. Zelevinsky. Discriminants,resultants, and multidimensional determinants. Mathematics: Theory & Applica-

tions. Birkhäuser, Boston, MA, 1994.

[29] Jonathan S. Golan. Power algebras over semirings, volume 488 of Mathemat-ics and its Applications. Kluwer Academic Publishers, Dordrecht, 1999. With

applications in mathematics and computer science.

[30] Nathan B. Grigg. Factorization of tropical polynomials in one and several vari-

ables. Brigham Young University Honors Thesis, 2007. Undergraduate Thesis.

[31] Walter Gubler. A guide to tropicalizations. Preprint, 2011.

[32] Kerstin Hept and Thorsten Theobald. Tropical bases by regular projections. Proc.Amer. Math. Soc., 137(7):2233–2241, 2009.

[33] Sven Herrmann, Anders Jensen, Michael Joswig, and Bernd Sturmfels. How to

draw tropical planes. Electron. J. Combin., 16(2, Special volume in honor of

Anders Bjorner):Research Paper 6, 26, 2009.

[34] Wilfried Imrich, J. M. S. Simões-Pereira, and Christina M. Zamfirescu. On op-

timal embeddings of metrics in graphs. J. Combin. Theory Ser. B, 36(1):1–15,

1984.

[35] Zur Izhakian. On the tropical rank of a tropical matrix. Communications inAlgebra, 37(11):3912âAS–3927, 2009.

[36] Zur Izhakian, Marianne Johnson, and Mark Kambites. Pure dimension and pro-

jectivity of tropical polytopes. Preprint, 2011.

[37] P. Jipsen and C. Tsinakis. A survey of residuated lattices. In Ordered algebraicstructures, volume 7 of Dev. Math., pages 19–56. Kluwer Acad. Publ., Dordrecht,

2002.

[38] M.M. Kapranov. A characterization of A-discriminantal hypersurfaces in terms

of logarithmic Gauss map. Math. Ann., 290(2):277–285, 1991.

[39] Eric Katz. Tropical geometry. Notes by Eric Katz of a course by Grigory Mikhalkin

at the University of Texas, February and March 2008.

148 BIBLIOGRAPHY

[40] Yusuke Kobayashi and Kazuo Murota. Induction of M-convex functions by link-

ing systems. Discrete Appl. Math., 155(11):1471–1480, 2007.

[41] Jack Koolen, Alice Lesser, and Vincent Moulton. Optimal realizations of generic

five-point metrics. European J. Combin., 30(5):1164–1171, 2009.

[42] Joseph P. S. Kung. Bimatroids and invariants. Adv. in Math., 30(3):238–249,

1978.

[43] G. L. Litvinov, V. P. Maslov, and G. B. Shpiz. Idempotent functional analysis. An

algebraic approach. Mat. Notes, 69, 2001.

[44] Dean Lucas. Weak maps of combinatorial geometries. Trans. Amer. Math. Soc.,206:247–279, 1975.

[45] Grigory Mikhalkin. Enumerative tropical algebraic geometry in R2. J. Amer.Math. Soc., 18(2):313–377, 2005.

[46] Mikhail Mitrofanov. Bimatroids and Gauss decomposition. European J. Combin.,28(4):1180–1195, 2007.

[47] K. Murota. Finding optimal minors of valuated bimatroids. Appl. Math. Lett.,8(4):37–41, 1995.

[48] Kazuo Murota. Convexity and Steinitz’s exchange property. Adv. Math.,124(2):272–311, 1996.

[49] Kazuo Murota and Akihisa Tamura. On circuit valuation of matroids. Adv. inAppl. Math., 26(3):192–225, 2001.

[50] James G. Oxley. Matroid theory. Oxford Science Publications. The Clarendon

Press Oxford University Press, New York, 1992.

[51] Sam Payne. Analytification is the limit of all tropicalizations. Math. Res. Lett.,16(3):543–556, 2009.

[52] Sam Payne. Fibers of tropicalization. Math. Z., 262(2):301–311, 2009.

[53] Jean-Eric Pin. Tropical semirings. In J. Gunawardena, editor, Idempotency. Cam-

bridge University Press, 1998.

[54] V. Puiseux. Recherches sur les fonctions algébriques. J. Math. Pures Appl.,15:365–480, 1850.

[55] Felipe Rincón. Computing tropical linear spaces, 2011. Preprint available from

http://arxiv.org/abs/1109.4130.

BIBLIOGRAPHY 149

[56] A. Schrijver. Matroids and linking systems. J. Combin. Theory Ser. B, 26(3):349–

369, 1979.

[57] Kristin M. Shaw. A tropical intersection product in matroidal fans. Preprint,

2011.

[58] David E. Speyer. Tropical Geometry. PhD thesis, University of California, Berke-

ley, 2005.

[59] David E. Speyer. Tropical linear spaces. SIAM J. Discrete Math., 22(4):1527–

1558, 2008.

[60] Stuart A. Steinberg. Lattice-ordered Rings and Modules. Springer, 2009.

[61] Bernd Sturmfels and Jenia Tevelev. Elimination theory for tropical varieties.

Mathematical Research Letters, pages 543–562, 2008.

[62] Bernd Sturmfels, Jenia Tevelev, and Josephine Yu. The Newton polytope of the

implicit equation. Mosc. Math. J., 7(2):327–346, 351, 2007.

[63] Bernd Sturmfels and Josephine Yu. Classification of six-point metrics. Electron.J. Combin., 11(1):Research Paper 44, 16 pp. (electronic), 2004.

[64] Bernd Sturmfels and Josephine Yu. Tropical implicitization and mixed fiber

polytopes. In Software for algebraic geometry, volume 148 of IMA Vol. Math.Appl., pages 111–131. Springer, New York, 2008.

[65] K. L. N. Swamy. Dually residuated lattice ordered semigroups. Math. Ann.,159:105–114, 1965.

[66] Bartel L. van der Waerden. Algebra. Zweiter Teil, volume 23 of HeidelbergerTaschenbücher. Springer Verlag, Berlin Heidelberg New York, fifth edition, 1967.

[67] Bartel L. van der Waerden. Einfürung in die algebraische Geometrie, volume 51

of Die Grundlehren der mathematischen Wissenschaften. Springer Verlag, Berlin

Heidelberg New York, second edition, 1973.

[68] Edouard Wagneur. Moduloïds and pseudomodules 1. dimension theory. DiscreteMathematics, 1998:57–73, 1991.

[69] Huaxiong Wang. Injective hulls of semimodules over additively-idempotent

semirings. Semigroup Forum, 48:377–379, 1994.

[70] Morgan Ward and R. P. Dilworth. Residuated lattices. Trans. Amer. Math. Soc.,45(3):335–354, 1939.

[71] Josephine Yu and Debbie Yuster. Representing tropical linear spaces by circuits.

In Formal Power Series and Algebraic Combinatorics, Proceedings, 2007.

Index

F -matroid, 64

F -valuated matroid, 64

F -weighted network, 108

rns-labeled graph, 135

absorption axiom, 38

additive ring valuation, 54

algebraic dual space, 48

algebraic torus, 21

Archimedean, 42

basis of a module, 57

Berkovich analytic space, 18

Boolean semiring, 42

canonical order, 40

classical matrix operations, 128

cofactor system, 105

column set, 105

combinatorial type, 68

commutative semiring, 38

complete, 12

complete intersection, 124

composition

of linking systems, 107

of morphisms, 121

contraction, 76

cycle, 99

degree-d homogenization, 27

dependent, 57

detour, 137

dimension, 12

dioids, 38

direct sum, 78

distance matrix, 128

divisible semifield, 8

division semiring, 38

domain set, 105

Dressian, 80

dual

algebraic, 48

orthogonal, 73

dual residuum, 43

dual space, 48

dually residuated

increasing map, 46

module, 43

elementary distance matrix, 130

elementary extension, 93

elementary matroidal extension function,

93, 98

equivalent projections, 12

extended graph, 92

extension, 92

elementary, 93

faithful, 39

free, 39

Gondran-Minoux dependent, 57

Gondran-Minoux pseudomorphism, 54

good section, 5

150

INDEX 151

gossip monoid, 129

gossip problem, 132

graph of a linking system, 104

idempotent, 38

idempotent module, 39

implicitization, 21

independent, 57

indicator, 52

induced order, 40

initial form, 5

irreducible, 57

kernel of a projection, 12

labeled graph, 135

labeled graph with detours, 137

weighted, 137

length, 131

linear functional, 48

linking, 108

linking system, 104

matroid over F , 64

matroid union, 79

matroidal extension function, 93

minimal, 135

minimal matroid, 65

minimal representative, 87

module, 38

monomial, 2

morphism, 113

natural order, 40

natural preorder, 39

Newton polytope, 10

optimal, 135

ordinary matrix operations, 128

orthogonal dual, 73

polynomial, 2

principal filter, 46

projection, 12

proportional, 64

pseudomorphisms of algebras, 54

pseudomorphisms of semirings, 54

Puiseux series, 2

range set, 105

rank, 57

real tropical semifield, 38

realizable valuated matroids, 65

realized by a graph, 135

reducible, 57

reflexive, 50

regular projection, 12

regular subdivision, 10

reparameterization, 26

represent a morphism, 114

representation matroid, 104

residual matroid, 65

residue field, 2

restriction

of a linking system, 105

of a matroid, 76

of a morphism, 117

row set, 105

semifield, 1, 38

semiring, 37

separated, 50

sink node, 108

source node, 108

support, 84

surplus length, 141

terminal node, 108

transpose, 49

trivial semifield, 2, 42

tropical basis

of a set, 5

of a variety, 7

152 INDEX

tropical identity matrix, 129

tropical linear monoid, 122

tropical linear space, 69

tropical prevariety

of a polynomial, 3

of a set, 4

tropical pseudomorphism, 54

tropical rational numbers, 38

tropical rational semifield, 38

tropical real number semifield, 1

tropical semifield, 2

tropical semiring, 42

tropical vanishing locus, 3

tropical variety, 5

tropicalization

of a variety, 7

of a polynomial, 3

of a rational function, 22

tropically dependent, 57

tropically unirational, 22

truncation, 72

underlying matroid, 65

valuated exchange condition, 64

valuated matroid, 64

on independent sets, 72

valuation ring, 2

vanish, 53

in the sense of Gondran-Minoux, 53

in the tropical sense, 53

weakly, 53

vanishing axiom, 55

vanishing sequence, 53

weakly dependent, 57

weight of a linking, 108

zero-sum free, 52

Summary

Tropical varieties, maps and

gossip

Tropical geometry is a relatively new field of mathematics that studies the tropicaliza-

tion map: a map that assigns a certain type of polyhedral complex, called a tropical

variety, to an embedded algebraic variety. In a sense, it translates algebraic geomet-

ric statements into combinatorial ones. An interesting feature of tropical geometry

is that there does not exist a good notion of morphism, or map, between tropical

varieties that makes the tropicalization map functorial. The main part of this thesis

studies maps between different classes of tropical varieties: tropical linear spaces and

tropicalizations of embedded unirational varieties.

The first chapter is a concise introduction to tropical geometry. It collects and

proves the main theorems. None of these results are new.

The second chapter deals with tropicalizations of embedded unirational varieties.

We give sufficient conditions on such varieties for there to exist a (not necessarily

injective) parametrization whose naive tropicalization is surjective onto the associated

tropical variety.

The third chapter gives an overview of the algebra related to tropical linear spaces.

Where fields and vector spaces are the central objects in linear algebra, so are semi-

fields and modules over semifields central to tropical linear algebra and the study

of tropical linear spaces. Most results in this chapter are known in some form, but

scattered among the available literature. The main purpose of this chapter is to col-

lect these results and to determine the algebraic conditions that suffice to give linear

algebra over the semifield a familiar feel. For example, under which conditions are

varieties cut out by linear polynomials closed under addition and scalar multiplica-

tion?

The fourth chapter comprises the biggest part of the thesis. The techniques used

are a combination of tropical linear algebra and matroid theory. Central objects are

153

154 INDEX

the valuated matroids introduced by Andreas Dress and Walter Wenzl. Among other

things the chapter contains a classification of functions on a tropical linear space

whose cycles are tropical linear subspaces, extending an old result on elementary

extensions of matroids by Henry Crapo. It uses Mikhalkin’s concept of a tropical mod-

ification to define the morphisms in a category whose objects are all tropical linear

spaces. Finally, we determine the structure of an open submonoid of the morphisms

from affine 2-space to itself as a polyhedral complex.

Finally, the fifth and last chapter is only indirectly related to maps. It studies a cer-

tain monoid contained in the tropicalization of the orthogonal group: the monoid that

is generated by the distance matrices under tropical matrix multiplication (i.e. where

addition is replaced by minimum, and multiplication by addition). This monoid gen-

eralizes a monoid that underlies the well-known gossip problem, to a setting where

information is transmitted only with a certain degree accuracy. We determine this

so-called gossip monoid for matrices up to size 4, and prove that in general it is a

polyhedral monoid of dimension equal to that of the orthogonal group.

Curriculum Vitae

Bartholomeus Johannes Frenk was born on February 3, 1981 in Schiedam, the Nether-

lands. After finishing VWO in 1993 at the Bisschop Bekkers College in Eindhoven, he

studied Applied Mathematics at the Eindhoven University of Technology. In 2006 he

graduated within the Discrete Algebra and Geometry group on algebras related to

knot theory. After working for nearly a year as a risk modelling analyst he started a

Ph.D. project in 2008 at the Eindhoven University of Technology, of which the results

are presented in this dissertation.

155

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