Adelic perturbation of rational functions and applications

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.