Adelic perturbation of rational functions and applications

TL;DR

This paper investigates the analytic continuation properties of power series derived from rational functions with almost quasi-polynomial coefficients, establishing a dichotomy that has implications for dynamical systems and number theory.

Contribution

It proves a Pólya-Carlson type dichotomy for these series under stability conditions, solving several conjectures in dynamical systems and number theory.

Findings

The series is either rational or has limited analytic continuation.

Application to the Artin-Mazur zeta function on abelian varieties.

Resolution of conjectures by Byszewski-Cornelissen and Bell-Miles-Ward.

Abstract

Let $\sum a_nx^n\in\bar{\mathbb{Q}}[[x]]$ be the power series representation of a rational function and let $f:\ \{0,1,\ldots\}\rightarrow \bar{\mathbb{Q}}$ be a so-called almost quasi-polynomial. Under a necessary stability condition, we prove that $\sum f(n)a_nx^n$ satisfies the P\'olya-Carlson dichotomy: it is either a rational function or it cannot be extended analytically to a strictly larger domain than its disk of convergence. This latter property is much stronger than being transcendental. The first application and motivation of our result is the solution of a conjecture by Byszewski-Cornelissen. This gives a complete understanding of the analytic continuation behavior of the Artin-Mazur zeta function associated to a dynamical system on an abelian variety. Further applications include the solution of a conjecture by Bell-Miles-Ward and a significant case of an open problem by Royals-Ward.