Algebras of Linear Growth
and
the Dynamical Mordell�Lang Conjecture
Dmitri [email protected]
Higher School of Economics,
Moscow, Russia
Spa, 2017
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 1 / 17
Graded algebras
We call an associative algebra A graded if
A = A0 ⊕A1 ⊕A2 ⊕ . . . ,
where A0 = k is a basic �eld, dim kAi <∞. All ouralgebras are graded.Hilbert function: hA(n) = dimAnHilbert series:HA(z) =
∑n≥0 z
ndimAn =∑
n≥0 znhA(n).
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 2 / 17
Graded algebras
We call an associative algebra A graded if
A = A0 ⊕A1 ⊕A2 ⊕ . . . ,
where A0 = k is a basic �eld, dim kAi <∞. All ouralgebras are graded.Hilbert function: hA(n) = dimAnHilbert series:HA(z) =
∑n≥0 z
ndimAn =∑
n≥0 znhA(n).
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 2 / 17
Graded algebras
We call an associative algebra A graded if
A = A0 ⊕A1 ⊕A2 ⊕ . . . ,
where A0 = k is a basic �eld, dim kAi <∞. All ouralgebras are graded.Hilbert function: hA(n) = dimAnHilbert series:HA(z) =
∑n≥0 z
ndimAn =∑
n≥0 znhA(n).
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 2 / 17
Graded algebras
We call an associative algebra A graded if
A = A0 ⊕A1 ⊕A2 ⊕ . . . ,
where A0 = k is a basic �eld, dim kAi <∞. All ouralgebras are graded.Hilbert function: hA(n) = dimAnHilbert series:HA(z) =
∑n≥0 z
ndimAn =∑
n≥0 znhA(n).
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 2 / 17
Rationality
An algebra A is called �nitely presented if it is de�nedby a �nite number of generators and relations.
Theorem (Govorov, 1972)
If the relations of a �nitely presented algebra A are
monomials in generators then HA(z) is a rational
function.
Conjecture (Govorov)
For each �nitely presented algebra A the Hilbert series
HA(z) is a rational function.
First counterexamples: Ufnarovski, 1978 and Shearer,1980.Open questions: Govorov conjecture for Noetherianalgebras and for Koszul algebras.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 3 / 17
Rationality
An algebra A is called �nitely presented if it is de�nedby a �nite number of generators and relations.
Theorem (Govorov, 1972)
If the relations of a �nitely presented algebra A are
monomials in generators then HA(z) is a rational
function.
Conjecture (Govorov)
For each �nitely presented algebra A the Hilbert series
HA(z) is a rational function.
First counterexamples: Ufnarovski, 1978 and Shearer,1980.Open questions: Govorov conjecture for Noetherianalgebras and for Koszul algebras.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 3 / 17
Rationality
An algebra A is called �nitely presented if it is de�nedby a �nite number of generators and relations.
Theorem (Govorov, 1972)
If the relations of a �nitely presented algebra A are
monomials in generators then HA(z) is a rational
function.
Conjecture (Govorov)
For each �nitely presented algebra A the Hilbert series
HA(z) is a rational function.
First counterexamples: Ufnarovski, 1978 and Shearer,1980.Open questions: Govorov conjecture for Noetherianalgebras and for Koszul algebras.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 3 / 17
Rationality
An algebra A is called �nitely presented if it is de�nedby a �nite number of generators and relations.
Theorem (Govorov, 1972)
If the relations of a �nitely presented algebra A are
monomials in generators then HA(z) is a rational
function.
Conjecture (Govorov)
For each �nitely presented algebra A the Hilbert series
HA(z) is a rational function.
First counterexamples: Ufnarovski, 1978 and Shearer,1980.Open questions: Govorov conjecture for Noetherianalgebras and for Koszul algebras.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 3 / 17
Rationality
An algebra A is called �nitely presented if it is de�nedby a �nite number of generators and relations.
Theorem (Govorov, 1972)
If the relations of a �nitely presented algebra A are
monomials in generators then HA(z) is a rational
function.
Conjecture (Govorov)
For each �nitely presented algebra A the Hilbert series
HA(z) is a rational function.
First counterexamples: Ufnarovski, 1978 and Shearer,1980.Open questions: Govorov conjecture for Noetherianalgebras and for Koszul algebras.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 3 / 17
Rationality in the linear growth caseAn algebra has linear growth, if GK�dimA ≤ 1, that is,for some c > 0 we have hA(n) = dimAn < c.
Example
Let A = 〈x, y|x2, yxy, xy2tx for all t ≥ 0〉. ThenAn = k{yn, xyn−1, yn−1x, xyn−2x} for n 6= 2t + 2 orAn = k{yn, xyn−1, yn−1x} otherwise.We have HA(z) = 1 + 2z + 4z2/(1− z)− z2
∑t≥0 z
2t .
Problem (Govorov conjecture for algebras of lineargrowth, GALG)
Suppose that an algebra A of linear growth is �nitely
presented. Is HA(z) a rational function?
For such algebras, HA(z) is rational i� hA(n) iseventually periodic, that is, ∃n0, T > 0 such thathA(n) = hA(n+ T ) for all n > n0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 4 / 17
Rationality in the linear growth caseAn algebra has linear growth, if GK�dimA ≤ 1, that is,for some c > 0 we have hA(n) = dimAn < c.
Example
Let A = 〈x, y|x2, yxy, xy2tx for all t ≥ 0〉. ThenAn = k{yn, xyn−1, yn−1x, xyn−2x} for n 6= 2t + 2 orAn = k{yn, xyn−1, yn−1x} otherwise.We have HA(z) = 1 + 2z + 4z2/(1− z)− z2
∑t≥0 z
2t .
Problem (Govorov conjecture for algebras of lineargrowth, GALG)
Suppose that an algebra A of linear growth is �nitely
presented. Is HA(z) a rational function?
For such algebras, HA(z) is rational i� hA(n) iseventually periodic, that is, ∃n0, T > 0 such thathA(n) = hA(n+ T ) for all n > n0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 4 / 17
Rationality in the linear growth caseAn algebra has linear growth, if GK�dimA ≤ 1, that is,for some c > 0 we have hA(n) = dimAn < c.
Example
Let A = 〈x, y|x2, yxy, xy2tx for all t ≥ 0〉. ThenAn = k{yn, xyn−1, yn−1x, xyn−2x} for n 6= 2t + 2 orAn = k{yn, xyn−1, yn−1x} otherwise.We have HA(z) = 1 + 2z + 4z2/(1− z)− z2
∑t≥0 z
2t .
Problem (Govorov conjecture for algebras of lineargrowth, GALG)
Suppose that an algebra A of linear growth is �nitely
presented. Is HA(z) a rational function?
For such algebras, HA(z) is rational i� hA(n) iseventually periodic, that is, ∃n0, T > 0 such thathA(n) = hA(n+ T ) for all n > n0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 4 / 17
Automaton algebras
Let X be a �nite generating set of an algebra A.Consider a multiplicative ordering `<' of the set of allwords in X. A word on X is called normal in A if it isnot a linear combination of less words. The set N of allnormal words is a linear basis of A.
De�nition (Ufnarovski)
An algebra A is called automaton if N is a regularlanguage.
Recall that a language is regular i� it is recognized by a�nite automaton.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 5 / 17
Automaton algebras
Let X be a �nite generating set of an algebra A.Consider a multiplicative ordering `<' of the set of allwords in X. A word on X is called normal in A if it isnot a linear combination of less words. The set N of allnormal words is a linear basis of A.
De�nition (Ufnarovski)
An algebra A is called automaton if N is a regularlanguage.
Recall that a language is regular i� it is recognized by a�nite automaton.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 5 / 17
Automaton algebras
Let X be a �nite generating set of an algebra A.Consider a multiplicative ordering `<' of the set of allwords in X. A word on X is called normal in A if it isnot a linear combination of less words. The set N of allnormal words is a linear basis of A.
De�nition (Ufnarovski)
An algebra A is called automaton if N is a regularlanguage.
Recall that a language is regular i� it is recognized by a�nite automaton.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 5 / 17
Automaton algebras
Let X be a �nite generating set of an algebra A.Consider a multiplicative ordering `<' of the set of allwords in X. A word on X is called normal in A if it isnot a linear combination of less words. The set N of allnormal words is a linear basis of A.
De�nition (Ufnarovski)
An algebra A is called automaton if N is a regularlanguage.
Recall that a language is regular i� it is recognized by a�nite automaton.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 5 / 17
Automaton algebras
Property (generalized Govorov theorem): If A is graded
automaton (that is, it is graded and automaton with
homogeneous X and a degree-compatible ordering `<'),
then HA(z) is a rational function.
Conjecture (Ufnarovski, 1990)
A �nitely presented algebra of linear growth is
automaton.
Conjecture (Ufnarovski conjecture for gradedalgebras, UGA)
A graded �nitely presented algebra of linear growth is
graded automaton.
UGA implies GALG.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 6 / 17
Automaton algebras
Property (generalized Govorov theorem): If A is graded
automaton (that is, it is graded and automaton with
homogeneous X and a degree-compatible ordering `<'),
then HA(z) is a rational function.
Conjecture (Ufnarovski, 1990)
A �nitely presented algebra of linear growth is
automaton.
Conjecture (Ufnarovski conjecture for gradedalgebras, UGA)
A graded �nitely presented algebra of linear growth is
graded automaton.
UGA implies GALG.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 6 / 17
Automaton algebras
Property (generalized Govorov theorem): If A is graded
automaton (that is, it is graded and automaton with
homogeneous X and a degree-compatible ordering `<'),
then HA(z) is a rational function.
Conjecture (Ufnarovski, 1990)
A �nitely presented algebra of linear growth is
automaton.
Conjecture (Ufnarovski conjecture for gradedalgebras, UGA)
A graded �nitely presented algebra of linear growth is
graded automaton.
UGA implies GALG.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 6 / 17
Automaton algebras
Property (generalized Govorov theorem): If A is graded
automaton (that is, it is graded and automaton with
homogeneous X and a degree-compatible ordering `<'),
then HA(z) is a rational function.
Conjecture (Ufnarovski, 1990)
A �nitely presented algebra of linear growth is
automaton.
Conjecture (Ufnarovski conjecture for gradedalgebras, UGA)
A graded �nitely presented algebra of linear growth is
graded automaton.
UGA implies GALG.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 6 / 17
The �nite characteristic case
Theorem
Suppose that the �eld k has a �nite characteristic. Then
both Govorov conjecture for algebra of linear growth and
Ufnarovski conjecture for graded algebras hold if and
only if k is an algebraic extension of its prime sub�eld.
`If' part: essentially, the case of �nite �eld.'Only if' part (counterexamples to GALG): later.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 7 / 17
The �nite characteristic case
Theorem
Suppose that the �eld k has a �nite characteristic. Then
both Govorov conjecture for algebra of linear growth and
Ufnarovski conjecture for graded algebras hold if and
only if k is an algebraic extension of its prime sub�eld.
`If' part: essentially, the case of �nite �eld.'Only if' part (counterexamples to GALG): later.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 7 / 17
The �nite characteristic case
Theorem
Suppose that the �eld k has a �nite characteristic. Then
both Govorov conjecture for algebra of linear growth and
Ufnarovski conjecture for graded algebras hold if and
only if k is an algebraic extension of its prime sub�eld.
`If' part: essentially, the case of �nite �eld.'Only if' part (counterexamples to GALG): later.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 7 / 17
The case of in�nite �eldWhat about the case char k = 0?
Example (Fermat algebras)
For α, β ∈ k×, let A = Aα,β be generated bya, b, c, x, y, z subject to 26 relations xc− αcx, yb− βcyand others. Then hA(n+ 3) is 10 or 11 according towhether the Fermat equality αn + βn = 1 holds. So, ithas no nonzero solution in k× for each n ≥ 3 if and onlyif hA(i) = 10 for all i ≥ 6 and each A = Aα,β .
Theorem
Let g ≥ 5 be an integer. If the �eld k is in�nite, then
there are in�nitely many (periodic) sequences hA for
g-generated quadratic k-algebras of linear growth. If If,in addition, k contains all primitive roots of unity, then
both the length d of the initial non-periodic segment
and the period T of hA can be arbitrary large.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 8 / 17
The case of in�nite �eldWhat about the case char k = 0?
Example (Fermat algebras)
For α, β ∈ k×, let A = Aα,β be generated bya, b, c, x, y, z subject to 26 relations xc− αcx, yb− βcyand others. Then hA(n+ 3) is 10 or 11 according towhether the Fermat equality αn + βn = 1 holds. So, ithas no nonzero solution in k× for each n ≥ 3 if and onlyif hA(i) = 10 for all i ≥ 6 and each A = Aα,β .
Theorem
Let g ≥ 5 be an integer. If the �eld k is in�nite, then
there are in�nitely many (periodic) sequences hA for
g-generated quadratic k-algebras of linear growth. If If,in addition, k contains all primitive roots of unity, then
both the length d of the initial non-periodic segment
and the period T of hA can be arbitrary large.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 8 / 17
The case of in�nite �eldWhat about the case char k = 0?
Example (Fermat algebras)
For α, β ∈ k×, let A = Aα,β be generated bya, b, c, x, y, z subject to 26 relations xc− αcx, yb− βcyand others. Then hA(n+ 3) is 10 or 11 according towhether the Fermat equality αn + βn = 1 holds. So, ithas no nonzero solution in k× for each n ≥ 3 if and onlyif hA(i) = 10 for all i ≥ 6 and each A = Aα,β .
Theorem
Let g ≥ 5 be an integer. If the �eld k is in�nite, then
there are in�nitely many (periodic) sequences hA for
g-generated quadratic k-algebras of linear growth. If If,in addition, k contains all primitive roots of unity, then
both the length d of the initial non-periodic segment
and the period T of hA can be arbitrary large.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 8 / 17
Skolem�Mahler�Lech theorem
Theorem (Skolem�Mahler�Lech)
If char k = 0 and an = c1an−1 + · · ·+ cdan−d is a linear
recurrence over k, then the zero set {n ≥ 0|an = 0} isthe �nite union of several arithmetic progressions and a
�nite set.
We refer to a set of nonnegative integers which is theunion of a �nite set and a �nite collection of arithmeticprogressions as SML.All known proofs essentially use p-adic analysis.
Example (Lech, 1953)
If the �eld k has characteristic p > 0 and t ∈ k istranscendental over the prime sub�eld Fp then thesequence an = (t+ 1)n − tn − 1 satis�es an = 0 i�n = pm with m ≥ 0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 9 / 17
Skolem�Mahler�Lech theorem
Theorem (Skolem�Mahler�Lech)
If char k = 0 and an = c1an−1 + · · ·+ cdan−d is a linear
recurrence over k, then the zero set {n ≥ 0|an = 0} isthe �nite union of several arithmetic progressions and a
�nite set.
We refer to a set of nonnegative integers which is theunion of a �nite set and a �nite collection of arithmeticprogressions as SML.All known proofs essentially use p-adic analysis.
Example (Lech, 1953)
If the �eld k has characteristic p > 0 and t ∈ k istranscendental over the prime sub�eld Fp then thesequence an = (t+ 1)n − tn − 1 satis�es an = 0 i�n = pm with m ≥ 0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 9 / 17
Skolem�Mahler�Lech theorem
Theorem (Skolem�Mahler�Lech)
If char k = 0 and an = c1an−1 + · · ·+ cdan−d is a linear
recurrence over k, then the zero set {n ≥ 0|an = 0} isthe �nite union of several arithmetic progressions and a
�nite set.
We refer to a set of nonnegative integers which is theunion of a �nite set and a �nite collection of arithmeticprogressions as SML.All known proofs essentially use p-adic analysis.
Example (Lech, 1953)
If the �eld k has characteristic p > 0 and t ∈ k istranscendental over the prime sub�eld Fp then thesequence an = (t+ 1)n − tn − 1 satis�es an = 0 i�n = pm with m ≥ 0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 9 / 17
Skolem�Mahler�Lech theorem
Theorem (Skolem�Mahler�Lech)
If char k = 0 and an = c1an−1 + · · ·+ cdan−d is a linear
recurrence over k, then the zero set {n ≥ 0|an = 0} isthe �nite union of several arithmetic progressions and a
�nite set.
We refer to a set of nonnegative integers which is theunion of a �nite set and a �nite collection of arithmeticprogressions as SML.All known proofs essentially use p-adic analysis.
Example (Lech, 1953)
If the �eld k has characteristic p > 0 and t ∈ k istranscendental over the prime sub�eld Fp then thesequence an = (t+ 1)n − tn − 1 satis�es an = 0 i�n = pm with m ≥ 0.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 9 / 17
The dynamical Mordell�Lang conjecture:
formulation
A challenging generalization:
Conjecture (The dynamical Mordell�Langconjecture)
Let V be a quasiprojective variety over k (of
characteristic zero), let Φ : V → V be any morphism,
and let α ∈ V. Then for each subvariety Y ⊂ V, the set
{n ≥ 0|Φn(α) ∈ Y } is SML.
See the book (which is unfortunately absent in ourlibrary...) :Jason P. Bell, Dragos Ghioca, Thomas J.Tucker, The Dynamical Mordell-lang Conjecture, AMS,2016 (Mathematical Surveys and Monographs, vol. 210)
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 10 / 17
The dynamical Mordell�Lang conjecture:
some results
Some important cases are known.
Theorem (Bell, 2006)
Let char k = 0. The conjecture is true provided that Vis an a�ne variety and Φ is a polynomial automorphism.
Note that this corollary implies theSkolem�Mahler�Lech theorem.
Theorem
For any �eld k, GALG implies the linear dynamical
Mordell�Lang conjecture.
So, we have implicationsUGA=⇒GALG=⇒Linear ML=⇒ Skolem�Mahler�Lech.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 11 / 17
The dynamical Mordell�Lang conjecture:
some results
Some important cases are known.
Theorem (Bell, 2006)
Let char k = 0. The conjecture is true provided that Vis an a�ne variety and Φ is a polynomial automorphism.
Note that this corollary implies theSkolem�Mahler�Lech theorem.
Theorem
For any �eld k, GALG implies the linear dynamical
Mordell�Lang conjecture.
So, we have implicationsUGA=⇒GALG=⇒Linear ML=⇒ Skolem�Mahler�Lech.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 11 / 17
The dynamical Mordell�Lang conjecture:
some results
Some important cases are known.
Theorem (Bell, 2006)
Let char k = 0. The conjecture is true provided that Vis an a�ne variety and Φ is a polynomial automorphism.
Note that this corollary implies theSkolem�Mahler�Lech theorem.
Theorem
For any �eld k, GALG implies the linear dynamical
Mordell�Lang conjecture.
So, we have implicationsUGA=⇒GALG=⇒Linear ML=⇒ Skolem�Mahler�Lech.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 11 / 17
The dynamical Mordell�Lang conjecture:
some results
Some important cases are known.
Theorem (Bell, 2006)
Let char k = 0. The conjecture is true provided that Vis an a�ne variety and Φ is a polynomial automorphism.
Note that this corollary implies theSkolem�Mahler�Lech theorem.
Theorem
For any �eld k, GALG implies the linear dynamical
Mordell�Lang conjecture.
So, we have implicationsUGA=⇒GALG=⇒Linear ML=⇒ Skolem�Mahler�Lech.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 11 / 17
Generalized dynamical Mordell�Lang
conjecture
Does dML implies UGA or (at least) GALG?Let A is generated and related in degrees at most D.Let V = mod-A≤D be the category of graded moduleswith generators and relations in degrees 0..D. ThenF : M 7→M≥1[1] is an endofunctor of V . For m > 0,let Vm = {M ∈ Ob V |dimM0 = m}.Then GALG is equivalent to the following categoricalversion of dML: Given a �nitely presented graded
algebra A of linear growth as above and a positive
integer m, is the set {n ≥ 0|Fn(A) ∈ Vm} always SML?
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 12 / 17
Generalized dynamical Mordell�Lang
conjecture
Does dML implies UGA or (at least) GALG?Let A is generated and related in degrees at most D.Let V = mod-A≤D be the category of graded moduleswith generators and relations in degrees 0..D. ThenF : M 7→M≥1[1] is an endofunctor of V . For m > 0,let Vm = {M ∈ Ob V |dimM0 = m}.Then GALG is equivalent to the following categoricalversion of dML: Given a �nitely presented graded
algebra A of linear growth as above and a positive
integer m, is the set {n ≥ 0|Fn(A) ∈ Vm} always SML?
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 12 / 17
Generalized dynamical Mordell�Lang
conjecture
Does dML implies UGA or (at least) GALG?Let A is generated and related in degrees at most D.Let V = mod-A≤D be the category of graded moduleswith generators and relations in degrees 0..D. ThenF : M 7→M≥1[1] is an endofunctor of V . For m > 0,let Vm = {M ∈ Ob V |dimM0 = m}.Then GALG is equivalent to the following categoricalversion of dML: Given a �nitely presented graded
algebra A of linear growth as above and a positive
integer m, is the set {n ≥ 0|Fn(A) ∈ Vm} always SML?
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 12 / 17
Thank you!
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 13 / 17
Proof of the main theorem`Only if' k an algebraic extension of Fp =⇒ UGA =⇒GALGFor example, let us show: k is �nite =⇒ GALGProof. There are �nite number (say, N) of isomorphismclasses of M ∈ mod-A≤D such that dimM0 ≤ C(where dimAn ≤ C for all n ≥ 0). Then the sequencehA(n) = dimFn(A)0 is periodic with both period andnon-periodic segment of length at most N .`If'k contains Fp[t] =⇒ GALG fails.Using Lech example (an = (t+ 1)n − tn − 1): there is a6-generated algebra with generators a, b, c, x, y, z withrelations xc− (t+ 1)cx, yc− tcy, zc− cz and 23 others(the same as for Fermat algebras). Then
hA(n+ 3) =
{11, n = pm for some m ≥ 0,10, otherwise.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 14 / 17
Proof of the main theorem`Only if' k an algebraic extension of Fp =⇒ UGA =⇒GALGFor example, let us show: k is �nite =⇒ GALGProof. There are �nite number (say, N) of isomorphismclasses of M ∈ mod-A≤D such that dimM0 ≤ C(where dimAn ≤ C for all n ≥ 0). Then the sequencehA(n) = dimFn(A)0 is periodic with both period andnon-periodic segment of length at most N .`If'k contains Fp[t] =⇒ GALG fails.Using Lech example (an = (t+ 1)n − tn − 1): there is a6-generated algebra with generators a, b, c, x, y, z withrelations xc− (t+ 1)cx, yc− tcy, zc− cz and 23 others(the same as for Fermat algebras). Then
hA(n+ 3) =
{11, n = pm for some m ≥ 0,10, otherwise.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 14 / 17
Proof of the main theorem`Only if' k an algebraic extension of Fp =⇒ UGA =⇒GALGFor example, let us show: k is �nite =⇒ GALGProof. There are �nite number (say, N) of isomorphismclasses of M ∈ mod-A≤D such that dimM0 ≤ C(where dimAn ≤ C for all n ≥ 0). Then the sequencehA(n) = dimFn(A)0 is periodic with both period andnon-periodic segment of length at most N .`If'k contains Fp[t] =⇒ GALG fails.Using Lech example (an = (t+ 1)n − tn − 1): there is a6-generated algebra with generators a, b, c, x, y, z withrelations xc− (t+ 1)cx, yc− tcy, zc− cz and 23 others(the same as for Fermat algebras). Then
hA(n+ 3) =
{11, n = pm for some m ≥ 0,10, otherwise.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 14 / 17
Proof of the main theorem`Only if' k an algebraic extension of Fp =⇒ UGA =⇒GALGFor example, let us show: k is �nite =⇒ GALGProof. There are �nite number (say, N) of isomorphismclasses of M ∈ mod-A≤D such that dimM0 ≤ C(where dimAn ≤ C for all n ≥ 0). Then the sequencehA(n) = dimFn(A)0 is periodic with both period andnon-periodic segment of length at most N .`If'k contains Fp[t] =⇒ GALG fails.Using Lech example (an = (t+ 1)n − tn − 1): there is a6-generated algebra with generators a, b, c, x, y, z withrelations xc− (t+ 1)cx, yc− tcy, zc− cz and 23 others(the same as for Fermat algebras). Then
hA(n+ 3) =
{11, n = pm for some m ≥ 0,10, otherwise.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 14 / 17
Proof of the main theorem`Only if' k an algebraic extension of Fp =⇒ UGA =⇒GALGFor example, let us show: k is �nite =⇒ GALGProof. There are �nite number (say, N) of isomorphismclasses of M ∈ mod-A≤D such that dimM0 ≤ C(where dimAn ≤ C for all n ≥ 0). Then the sequencehA(n) = dimFn(A)0 is periodic with both period andnon-periodic segment of length at most N .`If'k contains Fp[t] =⇒ GALG fails.Using Lech example (an = (t+ 1)n − tn − 1): there is a6-generated algebra with generators a, b, c, x, y, z withrelations xc− (t+ 1)cx, yc− tcy, zc− cz and 23 others(the same as for Fermat algebras). Then
hA(n+ 3) =
{11, n = pm for some m ≥ 0,10, otherwise.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 14 / 17
Normal words in algebras of linear growthThe next Proposition follows from [Belov and others,1997].
Proposition
If a (non-graded) algebra A of linear growth is
generated by a �nite set S, then there are U, V,W ⊂ S∗
such that each normal word in A has the form
w = acnb, where a ∈ U, b ∈ V, c ∈W,n ≥ 0.
Proposition
Suppose a language L consists of the words of the
above form. Then L is regular i� for each
a ∈ U, b ∈ V, c ∈W the set
Na,b,c = {n ∈ Z+|acnb ∈ L}
is SML.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 15 / 17
Normal words in algebras of linear growthThe next Proposition follows from [Belov and others,1997].
Proposition
If a (non-graded) algebra A of linear growth is
generated by a �nite set S, then there are U, V,W ⊂ S∗
such that each normal word in A has the form
w = acnb, where a ∈ U, b ∈ V, c ∈W,n ≥ 0.
Proposition
Suppose a language L consists of the words of the
above form. Then L is regular i� for each
a ∈ U, b ∈ V, c ∈W the set
Na,b,c = {n ∈ Z+|acnb ∈ L}
is SML.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 15 / 17
Normal words in algebras of linear growthThe next Proposition follows from [Belov and others,1997].
Proposition
If a (non-graded) algebra A of linear growth is
generated by a �nite set S, then there are U, V,W ⊂ S∗
such that each normal word in A has the form
w = acnb, where a ∈ U, b ∈ V, c ∈W,n ≥ 0.
Proposition
Suppose a language L consists of the words of the
above form. Then L is regular i� for each
a ∈ U, b ∈ V, c ∈W the set
Na,b,c = {n ∈ Z+|acnb ∈ L}
is SML.Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 15 / 17
Normal words in algebras of linear growth
Corollary
Suppose that the algebra A is automaton. Then there
are a generating set 1 ∈ S ⊂ A and an ordering such
that for some Q ⊂ S3 the set of normal words in A is
{acnb|n ≥ 0, (a, b, c) ∈ Q}.
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 16 / 17
Thank you very much!
Dmitri Piontkovski (HSE) Algebras of Linear Growth and the Dynamical Mordell�Lang ConjectureSpa, 2017 17 / 17