compactified and polyhedral versions of the amoeba … sadykov seminar 6 fe… · theorem....

COMPACTIFIED AND POLYHEDRAL VERSIONS

OF THE AMOEBA

OF AN ALGEBRAIC HYPERSURFACE

Timur Sadykov

Plekhanov Russian University, Moscow

Joint work with Mounir Nisse

to arrive here on April 8

DEFINITION. The Reinhardt diagram of a domain D ⊂ Cn is the

image of D under the map

(x1, . . . , xn) 7→ (|x1|, . . . , |xn|).

DEFINITION. The Reinhardt diagram of a domain D ⊂ Cn is the

image of D under the map

(x1, . . . , xn) 7→ (|x1|, . . . , |xn|).

DEFINITION. The amoeba Af of a Laurent polynomial f(x) (or ofthe algebraic hypersurface {f(x) = 0}) is defined to be the imageof the hypersurface f−1(0) under the map

Log : (x1, . . . , xn) 7→ (log |x1|, . . . , log |xn|).

TROPICAL GEOMETRY

TROPICAL GEOMETRY

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A tropical line

TROPICAL GEOMETRY

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A tropical line

A tropical polynomial is a concave, continuous, piecewise linearfunction.

TROPICAL GEOMETRY

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A tropical line

A tropical polynomial is a concave, continuous, piecewise linearfunction.

The set of points where a tropical polynomial is non-differentiableis called its associated tropical algebraic hypersurface.

Tropical line intersecting tropical cubic transversally

Pandanus Tectorius

Monstera

Monstera leaf: a closer look

EXAMPLE. The amoeba of a complex line.

p(x, y) = 1 + x + y

t t

t

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-2 -1 0 1 2

-2

-1

0

1

2

EXAMPLE.

p(x, y) = x + y + 6xy + x2y2

t

t

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THEOREM. (Forsberg, Passare, Tsikh, 2000.) Let p(x) be a Lau-rent polynomial and let {M} denote the family of connected com-ponents of the amoeba complement cAp(x). There exists an injectivefunction

ν : {M} → Zn ∩ Np(x)

such that the cone which is dual to Np(x) at the point ν(M) coincideswith the recession cone of M. In particular, the number of connectedcomponents of cAp(x) cannot be smaller than the number of verticesof Np(x) and cannot exceed the number of integer points in Np(x).

INVESTIGATING THE AMOEBA OF A POLYNOMIAL


1. What is the range of the map ν : {M} → Zn ∩ Np(x)?



2. Where in Rn is the amoeba Ap(x) located?



2. Where in Rn is the amoeba Ap(x) located?

3. For a given order v ∈ Zn ∩ Np(x), find a point x ∈ R

n whichbelongs to the connected component of M ⊂c Ap(x) with order v.

KNOWN RESULTS

KNOWN RESULTS

I.M. Gelfand, M.M. Kapranov, and A.V. Zelevinsky. Discriminants,resultants and multidimensional determinants. Birkhauser, 1994.

KNOWN RESULTS


T. Theobald. Computing amoebas, Experiment. Math. 11, no. 4(2002), 513-526.

KNOWN RESULTS



M. Passare and H. Rullgard. Amoebas, Monge-Ampre measures,and triangulations of the Newton polytope, Duke Math. J. 121,no. 3 (2004), 481-507.

KNOWN RESULTS




K. Purbhoo. A Nullstellensatz for amoebas, Duke Math. J. 141,no. 3 (2008), 407-445.

KNOWN RESULTS





Aron Lagerberg, Super currents and tropical geometry, Math. Z.270, no. 3-4 (2012), 1011-1050.

KNOWN RESULTS






T. Theobald and T. de Wolff. Approximating amoebas and coamoe-bas by sums of squares, Math. Comp. 84, no. 291 (2015), 455-473.

KNOWN RESULTS






T. Theobald and T. de Wolff. Approximating amoebas and coamoe-bas by sums of squares, Math. Comp. 84, no. 291 (2015), 455-473.

J. Forsgard, L.F. Matusevich, N. Mehlhop, and T. de Wolff. Lop-sided approximation of amoebas, arXiv:1608.08663v1.

COMPACTIFIED AMOEBAS

COMPACTIFIED AMOEBAS

DEFINITION 1. (Gelfand, Kapranov, Zelevinsky, 1994.) The com-pactified amoeba Af of a Laurent polynomial

f(x) =∑

s∈S

asxs

(or, equivalently, of the algebraic hypersurface {f(x) = 0}) is de-fined to be the image of the hypersurface f−1(0) under the momentmap

µS(x) :=

∑s∈S

s · |xs|

∑s∈S

|xs|.

Fig. 1. The affine and the compactified amoebas of the polynomialx + y + x2y2 + xy/2.

Fig. 2. The affine and the compactified amoebas of the polynomialx + y + x2y2 + 2xy

Fig. 3. The affine and the compactified amoeba of the polynomialx + 30xy + 20x2y + x3y + y2

Fig. 4. The affine and the compactified amoeba of the polynomialx+x2+y+xy3+x4y2+3x3y+10xy+10x2y+10xy2+15x2y2+10x3y2

WEIGHTED COMPACTIFIED AMOEBAS


The Newton polytope Np(x) of a Laurent polynomial p(x) is definedto be the convex hull in R

n of the support of p(x). We will oftendrop some of the subindices to simplify the notation.




DEFINITION. Following the ideas of Zharkov, we define the weightedmoment map associated with the algebraic hypersurface

{x ∈ Cn : f(x) :=

∑

s∈S

asxs = 0}

through

µf(x) :=

∑s∈S

s · |as||xs|

∑s∈S

|as||xs|.




DEFINITION. Following the ideas of Zharkov, we define the weightedmoment map associated with the algebraic hypersurface

{x ∈ Cn : f(x) :=

∑

s∈S

asxs = 0}

through

µf(x) :=

∑s∈S

s · |as||xs|

∑s∈S

|as||xs|.

It follows from the general theory of moment maps that

µf(Cn) ⊆ Nf .

DEFINITION. By the weighted compactified amoeba of an alge-braic hypersurface

H = {x ∈ Cn : f(x) = 0}

we will mean the set µf(H). We denote it by WCA(f).

DEFINITION. By the weighted compactified amoeba of an alge-braic hypersurface

H = {x ∈ Cn : f(x) = 0}

we will mean the set µf(H). We denote it by WCA(f).

Recall that the Hadamard power of order r ∈ R of a polynomialf(x) =

∑s∈S

asxs is defined to be f [r](x) :=

∑s∈S

arsx

s.

THEOREM 2. Let f be a polynomial in C[x±11 , · · · , x±1

n ] with theNewton polytope N such that |aα| ≥ 1 for every α ∈ Vert(N ).Assume that the function which assigns to each α ∈ N ∩Z

n the realnumber log |aα| is concave, and the subdivision of N dual to thetropical hypersurface Γ associated to the tropical polynomial ftrop

defined by:ftrop(ζ) = max

α∈N∩Zn{log |aα| + 〈α, ζ〉}

is a triangulation. Then the set-theoretical limit

P∞f := lim

r→∞WCA(f [r]) (1)

is a polyhedral complex. Besides, if n = 2 then P∞f is a simplicial

complex.Moreover, its complement in N has the same topology of the com-

plement of the amoeba A of f , i.e.

π0(Rn \ A) = π0(N \ P∞

f ).

The connected components of the complement of P∞f in the New-

ton polytope are not necessarily convex. The amoeba, the com-pactified amoeba and the associated polyhedral complex for thepolynomial x + y + x2y2 + cxy are depicted in Figures and forc = 1/2 and c = 2, respectively.

Fig. 5. The affine and the compactified amoebas of the polynomialx + y + x2y2 + xy/2. The polyhedral complex coincides with thecompactified amoeba of this polynomial.

Fig. 6. The affine amoeba, the compactified amoeba and the poly-hedral complex of the polynomial x + y + x2y2 + 2xy

The amoeba, the compactified amoeba and the associated polyhe-dral complex for the polynomial x+y+xy2+x2y+cxy are depictedin Figures 7 and 8 for c = 1/2 and c = 5, and respectively.

Fig. 7. The affine amoeba, the compactified amoeba and the poly-hedral complex of the polynomial x + y + xy2 + x2y + xy/2

Fig. 8. The affine amoeba, the compactified amoeba and the poly-hedral complex of the polynomial x + y + xy2 + x2y + 5xy

Fig. 9. The affine amoeba, the compactified amoeba and the poly-hedral complex of the polynomial x + 30xy + 20x2y + x3y + y2

Fig. 10. The affine amoeba, the compactifiedamoeba and the polyhedral complex of the polynomial1 + 3x + 3y + x2y + 4x3y + xy2 + 10x2y2 + 4xy3

Fig. 11. The affine amoeba, the compactifiedamoeba, the weighted compactified amoebas of the1st, 2nd and 3rd Hadamard powers of the polynomialx+x2+y+xy3+x4y2+3x3y+10xy+10x2y+10xy2+15x2y2+10x3y2,and a bounded component of its deformation vanishing at the latticepoint (2, 2)

OTHER COMBINATORIAL OBJECTS

ASSOCIATED WITH AN ALGEBRAIC HYPERSURFACE




G. Mikhalkin. Decomposition into pairs-of-pants for complex al-gebraic hypersurfaces, Topology 43, no. 5, (2004), 1035-1065.

CONJECTURE 3. Let f(x1, . . . , xn) ∈ C[x±11 , . . . , x±1

1 ] be a Lau-rent polynomial. Denote by Af ⊂ Nf its compactified amoebaand by {M} the set of (nonempty) connected components of thecomplement of Af in the Newton polytope Nf .We furthermore denote by ν(M) ∈ Nf ∩ Z

n the order of such acomponent.

There exists a polyhedral complex Pf with the following proper-ties:

1. Pf ⊂ Nf .

2. The polyhedral complex Pf is a deformation retract of thecompactified amoeba Af .

3. For any complement component M of Nf \ Af the only integerpoint that belongs to this component is its order: M ∩ Z

n = ν(M).

OPEN QUESTIONS

OPEN QUESTIONS

1. Give explicit analytic formula for Pf which works for all f .

OPEN QUESTIONS


2. Describe all f such that Pf is a simplicial complex.

OPEN QUESTIONS



3. What are the vertices of the polyhedral complex Pf?

OPEN QUESTIONS




4. What is the volume of Pf?

OPEN QUESTIONS




4. What is the volume of Pf?

5. How different can the volumes of the connected components ofNf \ Pf be?

A NEW ONLINE TOOL FOR AUTOMATED GENERATION

OF MATLAB CODE FOR DEPICTING AMOEBAS:

http://dvbogdanov.ru/?page=amoeba

PICTURES IN THE PRESENTATION HAVE BEEN GENERATED

BY MEANS OF THIS WEB-SERVICE

PREPRINT AVAILABLE AT

https://www.researchgate.net/profile/Timur Sadykov

compactified and polyhedral versions of the amoeba … sadykov seminar 6 fe… · theorem....

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