Vol. 15, No. 7, 2021

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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

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 an := f(φn(x0)), where φ : X −−→ X and f : X −−→ 1 are rational maps defined over K and x0 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 an G is a finite union of arithmetic progressions along with a set of upper Banach density zero. In addition, we show that if an 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 𝔾md and maps Ψ : 𝔾md 𝔾md and g : 𝔾md 𝔾m such that an = (g Ψn)(y) for some y 𝔾md. We then obtain results about the coefficients of D-finite power series using these facts.

$S$-units, $D$-finite series, arithmetic dynamics, algebraic groups, orbits
Mathematical Subject Classification
Primary: 14E05, 37P55
Secondary: 12H05
Received: 18 May 2020
Revised: 6 January 2021
Accepted: 4 February 2021
Published: 1 November 2021
Jason P. Bell
Department of Pure Mathematics
University of Waterloo
Waterloo, ON
Shaoshi Chen
Key Laboratory of Mathematics Mechanization
Academy of Mathematics and Systems Science
Chinese Academy of Sciences
Ehsaan Hossain
Department of Mathematical & Computational Sciences
University of Toronto Mississauga
Mississauga, ON