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Jason Bell

Publications and source records attributed to Jason Bell.

At least 37 records · Page 2Linked to original sources
arXiv2022

Rational self-maps with a regular iterate on a semiabelian variety

Let $G$ be a semiabelian variety defined over an algebraically closed field $K$ of characteristic $0$. Let $Φ\colon G\dashrightarrow G$ be a dominant rational self-map. Assume that an iterate $Φ^m \colon G \to G$ is regular for some $m \geqslant 1$ and that there exists no non-constant homomorphism $τ: G\to G_0$ of semiabelian varieties such that $τ\circ Φ^{m k}=τ$ for some $k \geqslant 1$. We show that under these assumptions $Φ$ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp.

math.NT↗
arXiv2022

Topological invariants for words of linear factor complexity

Given a finite alphabet $Σ$ and a right-infinite word $w$ over the alphabet $Σ$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work up to an equivalence in which two words are equivalent if they have the exact same set of factors (finite contiguous subwords). We show that ${\rm Rec}(w)$ can be endowed with a natural topology and we show that if $w$ is word of linear factor complexity then ${\rm Rec}(w)$ is a finite topological space. In addition, we note that there are examples which show that if $f:\mathbb{N}\to \mathbb{N}$ is a function that tends to infinity as $n\to \infty$ then there is a word whose factor complexity function is ${\rm O}(nf(n))$ such that ${\rm Rec}(w)$ is an infinite set. Finally, we pose a realization problem: which finite topological spaces can arise as ${\rm Rec}(w)$ for a word of linear factor complexity?

cs.FL↗
arXiv2022

A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let $K$ be an algebraically closed field of positive characteristic, let $G$ be a finitely generated subgroup of the multiplicative group of $K$, and let $X$ be a (irreducible) quasiprojective variety defined over $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\rightarrow X$ and $f\colon X\rightarrow\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. We show that the set of $n$ for which $a_n\in G$ is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and the $φ$ orbit of $x$ is Zariski dense in $X$ then {there is} a multiplicative torus $\mathbb{G}_m^d$ and maps $Ψ:\mathbb{G}_m^d \to \mathbb{G}_m^d$ and $g:\mathbb{G}_m^d \to \mathbb{G}_m$ such that $a_n = g\circ Ψ^n(y)$ for some $y\in \mathbb{G}_m^d$. We then describe various applications of our results.

math.NT↗
arXiv2022

A conjecture strengthening the Zariski dense orbit problem for birational maps of dynamical degree one

We formulate a strengthening of the Zariski dense orbit conjecture for birational maps of dynamical degree one. So, given a quasiprojective variety $X$ defined over an algebraically closed field $K$ of characteristic $0$, endowed with a birational self-map $ϕ$ of dynamical degree $1$, we expect that either there exists a non-constant rational function $f:X\dashrightarrow \mathbb{P}^1$ such that $f\circ ϕ=f$, or there exists a proper subvariety $Y\subset X$ with the property that for any invariant proper subvariety $Z\subset X$, we have that $Z\subseteq Y$. We prove our conjecture for automorphisms $ϕ$ of dynamical degree $1$ of semiabelian varieties $X$. Also, we prove a related result for regular dominant self-maps $ϕ$ of semiabelian varieties $X$: assuming $ϕ$ does not preserve a non-constant rational function, we have that the dynamical degree of $ϕ$ is larger than $1$ if and only if the union of all $ϕ$-invariant proper subvarieties of $X$ is Zariski dense. We give applications of our results to representation theoretic questions about twisted homogeneous coordinate rings associated to abelian varieties.

math.DS↗
arXiv2021

A height gap theorem for coefficients of Mahler functions

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are \emph{gaps} in the possible growths. In proving this height gap theorem, we obtain that a $k$-Mahler function is $k$-regular if and only if its coefficients have height in $O(\log n)$. Furthermore, we deduce that, over an arbitrary ground field of characteristic zero, a $k$-Mahler function is $k$-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas.

math.NT↗
arXiv2021

Cogrowth Series for Free Products of Finite Groups

Given a finitely generated group with generating set $S$, we study the cogrowth sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when $G$ has a finite-index free subgroup (using a result of Dunwoody). In this work we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if $S$ is a finite symmetric generating set for a group $G$ and if $a_n$ denotes the number of words of length $n$ over the alphabet $S$ that are equal to $1$ then $\limsup_n a_n^{1/n}$ exists and is either $1$, $2$, or at least $2\sqrt{2}$.

math.CO↗
arXiv2021

Automatic Sequences of Rank Two

Given a right-infinite word $\bf x$ over a finite alphabet $A$, the rank of $\bf x$ is the size of the smallest set $S$ of words over $A$ such that $\bf x$ can be realized as an infinite concatenation of words in $S$. We show that the property of having rank two is decidable for the class of $k$-automatic words for each integer $k\ge 2$.

cs.FL↗
arXiv2021

Noncommutative rational Pólya series

A (noncommutative) Pólya series over a field $K$ is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of $K^\times$. We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton.

math.CO↗
arXiv2020

Promoting circular-orderability to left-orderability

Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group $G$ is left-orderable if and only if $G \times \mathbb{Z}/n\mathbb{Z}$ is circularly-orderable for all $n > 1$. This implies that every circularly-orderable group which is not left-orderable gives rise to a collection of positive integers that exactly encode the obstruction to left-orderability, which we call the obstruction spectrum. We precisely describe the behaviour of the obstruction spectrum with respect to torsion, and show that this same behaviour can be mirrored by torsion-free groups, whose obstruction spectra are in general more complex.

math.GR↗
arXiv2020

Mahler's and Koksma's classifications in fields of power series

Let $q$ a prime power and ${\mathbb F}_q$ the finite field of $q$ elements. We study the analogues of Mahler's and Koksma's classifications of complex numbers for power series in ${\mathbb F}_q((T^{-1}))$. Among other results, we establish that both classifications coincide, thereby answering a question of Ooto.

math.NT↗
arXiv2020

Invariant hypersurfaces

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$-varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $ϕ_1,ϕ_2: Z \to X$ are dominant rational maps from a (possibly nonreduced) irreducible scheme $Z$ of finite-type to an algebraic variety $X$, with the property that there are infinitely many hypersurfaces on $X$ whose scheme-theoretic inverse images under $ϕ_1$ and $ϕ_2$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $gϕ_1=gϕ_2$. In the case when $Z$ is also reduced the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou-Hrushovski theorem to generalised algebraic $\mathcal D$-varieties and of Cantat's theorem to self-correspondences.

math.AG↗
arXiv2019

On the Complexity of the Cogrowth Sequence

Given a finitely generated group with generating set $S$, we study the \emph{cogrowth} sequence, which is the number of words of length $n$ over the alphabet $S$ that are equal to one. This is related to the probability of return for walks in a Cayley graph with steps from $S$. We prove that the cogrowth sequence is not $P$-recursive when~$G$ is an amenable group of superpolynomial growth, answering a question of Garrabant and Pak.

math.CO↗
arXiv2019

Effective versions of two theorems of Rado

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.

math.CO↗
arXiv2019

F-sets and finite automata

The classical notion of a k-automatic subset of the natural numbers is here extended to that of an F-automatic subset of an arbitrary finitely generated abelian group $Γ$ equipped with an arbitrary endomorphism F. This is applied to the isotrivial positive characteristic Mordell-Lang context where F is the Frobenius action on a commutative algebraic group G over a finite field, and $Γ$ is a finitely generated F-invariant subgroup of G. It is shown that the F-subsets of $Γ$ introduced by the second author and Scanlon are F-automatic. It follows that when G is semiabelian and X is a closed subvariety then X intersect $Γ$ is F-automatic. Derksen's notion of a k-normal subset of the natural numbers is also here extended to the above abstract setting, and it is shown that F-subsets are F-normal. In particular, the X intersect $Γ$ appearing in the Mordell-Lang problem are F-normal. This generalises Derksen's Skolem-Mahler-Lech theorem to the Mordell-Lang context.

math.NT↗
arXiv2019

Dynamical Uniform Bounds for Fibers and a Gap Conjecture

We prove a uniform version of the Dynamical Mordell-Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our first result, we assume $X$ is a quasi-projective variety defined over a field $K$ of characteristic $0$, endowed with the action of an étale endomorphism $Φ$, and $f\colon X\to Y$ is a morphism with $Y$ a quasi-projective variety defined over $K$. Then for any $x\in X(K)$, if for each $y\in Y(K)$, the set $S_y:=\{n\in \mathbb{N}\colon f(Φ^n(x))=y\}$ is finite, then there exists a positive integer $N$ such that $\#S_y\le N$ for each $y\in Y(K)$. For our second result, we let $K$ be a number field, $f:X\dashrightarrow \mathbb{P}^1$ is a rational map, and $Φ$ is an arbitrary endomorphism of $X$. If $\mathcal{O}_Φ(x)$ denotes the forward orbit of $x$ under the action of $Φ$, then either $f(\mathcal{O}_Φ(x))$ is finite, or $\limsup_{n\to\infty} h(f(Φ^n(x)))/\log(n)>0$, where $h(\cdot)$ represents the usual logarithmic Weil height for algebraic points.

math.NT↗
arXiv2018

Becker's conjecture on Mahler functions

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^γQ(z)$ for some explicit non-negative integer $γ$ and such that $1/Q(z)$ is $k$-regular.

math.NT↗
arXiv2018

The Dixmier-Moeglin equivalence, Morita equivalence, and homeomorphism of spectra

Let $k$ be a field and let $R$ be a left noetherian $k$-algebra. The algebra $R$ satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the ${\rm Spec}(R)$ and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if $R$ and $S$ are two left noetherian $k$-algebras with ${\rm dim}_k(R), {\rm dim}_k(S)<|k|$ then if $R$ and $S$ have homeomorphic spectra then $R$ satisfies the Dixmier-Moeglin equivalence if and only if $S$ does. In particular, the topology of ${\rm Spec}(R)$ can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if $k$ is uncountable and $R$ is affine noetherian and its prime spectrum is a disjoint union of subspaces that are each homeomorphic to the spectrum of an affine commutative ring then $R$ satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if $k$ is countable and $R$ is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that $R$ and $S$ are left noetherian $k$-algebras that satisfy the Dixmier-Moeglin equivalence then $R\otimes_k S$ does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that $eRe$ also satisfies the Dixmier-Moeglin equivalence, where $e$ is a nonzero idempotent of $R$.

math.RA↗