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

Bell, Jason P.; Chen, Shaoshi; Hossain, Ehsaan

doi:10.2140/ant.2021.15.1699

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

Jason P. Bell, Shaoshi Chen and Ehsaan Hossain

Vol. 15 (2021), No. 7, 1699–1728

DOI: 10.2140/ant.2021.15.1699

Abstract

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finitely generated subgroup of the multiplicative group of $K$ , and let $X$ be a quasiprojective variety defined over $K$ . We consider $K$ -valued sequences of the form $a_{n} : = f (φ^{n} (x_{0}))$ , where $φ : X --\to X$ and $f : X --\to ℙ^{1}$ 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 upper 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 is a multiplicative torus $𝔾_{m}^{d}$ and maps $Ψ : 𝔾_{m}^{d} \to 𝔾_{m}^{d}$ and $g : 𝔾_{m}^{d} \to 𝔾_{m}$ such that $a_{n} = (g \circ Ψ^{n}) (y)$ for some $y \in 𝔾_{m}^{d}$ . We then obtain results about the coefficients of $D$ -finite power series using these facts.

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Keywords

$S$-units, $D$-finite series, arithmetic dynamics, algebraic groups, orbits

Mathematical Subject Classification

Primary: 14E05, 37P55

Secondary: 12H05

Milestones

Received: 18 May 2020

Revised: 6 January 2021

Accepted: 4 February 2021

Published: 1 November 2021

Authors

Jason P. Bell
Department of Pure Mathematics University of Waterloo Waterloo, ON Canada

Shaoshi Chen
Key Laboratory of Mathematics Mechanization Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

Ehsaan Hossain
Department of Mathematical & Computational Sciences University of Toronto Mississauga Mississauga, ON Canada

Vol. 15, No. 7, 2021