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Jason P. Bell

Publications and source records attributed to Jason P. Bell.

At least 37 records · Page 2Linked to original sources
arXiv2021

On Dynamical Cancellation

Let $X$ be a projective variety and let $f$ be a dominant endomorphism of $X$, both of which are defined over a number field $K$. We consider a question of the second author, Meng, Shibata, and Zhang, which asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots$ eventually stabilizes, where $Y\subset X$ is a subvariety invariant under $f$. We show this question has an affirmative answer when the map $f$ is étale. We also look at a related problem of showing that there is some integer $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geq 0$, we necessarily have $f^{s_{0}}(x) = f^{s_{0}}(y)$. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on $\mathbb{P}^1$ where we allow for composition by multiple different maps $f_1,\dots,f_r$.

math.AG↗
arXiv2021

The upper density of an automatic set is rational

Given a natural number $k\ge 2$ and a $k$-automatic set $S$ of natural numbers, we show that the lower density and upper density of $S$ are recursively computable rational numbers and we provide an algorithm for computing these quantities. In addition, we show that for every natural number $k\ge 2$ and every pair of rational numbers $(α,β)$ with $0<α<β<1$ or with $(α,β)\in \{(0,0),(1,1)\}$ there is a $k$-automatic subset of the natural numbers whose lower density and upper density are $α$ and $β$ respectively, and we show that these are precisely the values that can occur as the lower and upper densities of an automatic set.

cs.FL↗
arXiv2021

A Tits alternative for rational functions

We prove an analog of the Tits alternative for rational functions. In particular, we show that if $S$ is a finitely generated semigroup of rational functions over the complex numbers, then either $S$ has polynomially bounded growth or $S$ contains a nonabelian free semigroup. We also show that if f and g are polarizable maps over any field that do not have the same set of preperiodic points, then the semigroup generated by f and g contains a nonabelian free semigroup.

math.NT↗
arXiv2021

Lie complexity of words

Given a finite alphabet $Σ$ and a right-infinite word $\bf w$ over $Σ$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to \mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under cyclic shift) of length-$n$ factors $x$ of $\bf w$ with the property that every element of the conjugacy class appears in $\bf w$. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity, and as a result we show that words of linear factor complexity have at most finitely many primitive factors $y$ with the property that $y^n$ is again a factor for every $n$. We then look at automatic sequences and show that the Lie complexity function of a $k$-automatic sequence is again $k$-automatic.

cs.FL↗
arXiv2020

On the importance of being primitive

We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings.

math.RA↗
arXiv2020

Rational dynamical systems, $S$-units, and $D$-finite power series

Let $K$ be an algebraically closed field of characteristic zero and let $G$ be a finitely generated subgroup of the multiplicative group of $K$. We consider $K$-valued sequences of the form $a_n:=f(φ^n(x_0))$, where $φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $K$ and $x_0\in X$ is a point whose forward orbit avoids the indeterminacy loci of $φ$ and $f$. Many classical sequences from number theory and algebraic combinatorics fall under this dynamical framework, and 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 Banach density zero. In addition, we show that if $a_n\in G$ for every $n$ and $X$ is irreducible and the $φ$ orbit of $x$ is Zariski dense in $X$ then there are 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 obtain results about the coefficients of $D$-finite power series using these facts.

math.NT↗
arXiv2020

Height Gap Conjectures, $D$-Finiteness, and Weak Dynamical Mordell-Lang

In previous work, the first author, Ghioca, and the third author introduced a broad dynamical framework giving rise to many classical sequences from number theory and algebraic combinatorics. Specifically, these are sequences of the form $f(Φ^n(x))$, where $Φ\colon X\to X$ and $f\colon X\to\mathbb{P}^1$ are rational maps defined over $\overline{\mathbb{Q}}$ and $x\in X(\overline{\mathbb{Q}})$ is a point whose forward orbit avoids the indeterminacy loci of $Φ$ and $f$. They conjectured that if the sequence is infinite, then $\limsup \frac{h(f(Φ^n(x)))}{\log n} > 0$. They also made a corresponding conjecture for $\liminf$ and showed that it implies the Dynamical Mordell-Lang Conjecture. In this paper, we prove the $\limsup$ conjecture as well as the $\liminf$ conjecture away from a set of density $0$. As applications, we prove results concerning the growth rate of coefficients of $D$-finite power series as well as the Dynamical Mordell-Lang Conjecture up to a set of density $0$.

math.NT↗
arXiv2019

A refinement of Christol's theorem for algebraic power series

A famous result of Christol gives that a power series $F(t)=\sum_{n\ge 0} f(n)t^n$ with coefficients in a finite field $\mathbb{F}_q$ of characteristic $p$ is algebraic over the field of rational functions in $t$ if and only if there is a finite-state automaton accepting the base-$p$ digits of $n$ as input and giving $f(n)$ as output for every $n\ge 0$. An extension of Christol's theorem, giving a complete description of the algebraic closure of $\mathbb{F}_q(t)$, was later given by Kedlaya. When one looks at the support of an algebraic power series, that is the set of $n$ for which $f(n)\neq 0$, a well-known dichotomy for sets generated by finite-state automata shows that the support set is either sparse---with the number of $n\le x$ for which $f(n)\neq 0$ bounded by a polynomial in $\log(x)$---or it is reasonably large in the sense that the number of $n\le x$ with $f(n)\neq 0$ grows faster than $x^α$ for some positive $α$. The collection of algebraic power series with sparse supports forms a ring and we give a purely algebraic characterization of this ring in terms of Artin-Schreier extensions and we extend this to the context of Kedlaya's work on generalized power series.

math.NT↗
arXiv2019

An analogue of Ruzsa's conjecture for polynomials over finite fields

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in\mathbb{N}$ and $f(n)=O(θ^n)$ with $θ<e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field.

math.NT↗
arXiv2019

Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer

Let $k$ be a field and let $A$ be a finitely generated $k$-algebra. The algebra $A$ is said to be cancellative if whenever $B$ is another $k$-algebra with the property that $A[x]\cong B[x]$ then we necessarily have $A\cong B$. An important result of Abhyankar, Eakin, and Heinzer shows that if $A$ is a finitely generated commutative integral domain of Krull dimension one then it is cancellative. We consider the question of cancellation for finitely generated not-necessarily-commutative domains of Gelfand-Kirillov dimension one, and show that such algebras are necessarily cancellative when the characteristic of the base field is zero. In particular, this recovers the cancellation result of Abhyankar, Eakin, and Heinzer in characteristic zero when one restricts to the commutative case. We also provide examples that show affine domains of Gelfand-Kirillov dimension one need not be cancellative when the base field has positive characteristic, giving a counterexample to a conjecture of Tang, the fourth-named author, and Zhang. In addition, we prove a skew analogue of the result of Abhyankar-Eakin-Heinzer, in which one works with skew polynomial extensions as opposed to ordinary polynomial rings.

math.RA↗
arXiv2019

D-finiteness, rationality, and height

Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic 0 is D-finite and its coefficients belong to a finite set then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series `look like' the coefficients of a rational function then the power series is rational. Our work relies on the theory of Weil heights, the Manin-Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences.

math.NT↗
arXiv2018

On free subgroups in division rings

Let $K$ be a field and let $σ$ be an automorphism and let $δ$ be a $σ$-derivation of $K$. Then we show that the multiplicative group of nonzero elements of the division ring $D=K(x;σ,δ)$ contains a free non-cyclic subgroup unless $D$ is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free non-abelian solvable-by-finite groups always contain free non-cyclic subgroups.

math.RA↗
arXiv2018

Periodic subvarieties of semiabelian varieties and annihilators of irreducible representations

Let $G$ be a semiabelian variety defined over a field of characteristic $0$, endowed with an endomorphism $Φ$. We prove there is no proper subvariety $Y\subset G$ which intersects the orbit of each periodic point of $G$ under the action of $Φ$. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras.

math.RA↗
arXiv2017

Bounding periods of subvarieties of (P^1)^n

Using methods of p-adic analysis, along with the powerful result of Medvedev-Scanlon (Annals of Mathematics, 2014) for the classification of periodic subvarieties of (P^1)^n, we bound the length of the orbit of a periodic subvariety Y of (P^1)^n under the action of a dominant endomorphism.

math.NT↗
arXiv2017

On a Dynamical Mordell-Lang Conjecture for Coherent Sheaves

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman's theorem that we prove in the appendix.

math.AG↗
arXiv2017

Free Subalgebras of Graded Algebras

Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.

math.RA↗
arXiv2017

Congruences modulo cyclotomic polynomials and algebraic independence for $q$-series

We prove congruence relations modulo cyclotomic polynomials for multisums of $q$-factorial ratios, therefore generalizing many well-known $p$-Lucas congruences. Such congruences connect various classical generating series to their $q$-analogs. Using this, we prove a propagation phenomenon: when these generating series are algebraically independent, this is also the case for their $q$-analogs.

math.CO↗