A general criterion for the P\'{o}lya-Carlson dichotomy and application

TL;DR

This paper establishes a broad criterion for when irrational power series have the unit circle as a natural boundary, and applies it to characterize the algebraic nature of Artin-Mazur zeta functions, revealing their boundary behavior in the transcendental case.

Contribution

It provides a general criterion for natural boundaries of power series over number fields and characterizes the algebraic and transcendental cases of Artin-Mazur zeta functions.

Findings

Power series with coefficients in a number field can have the unit circle as a natural boundary.

The Artin-Mazur zeta function is algebraic or transcendental depending on the matrix A.

In the transcendental case, the zeta function admits the circle of convergence as a natural boundary.

Abstract

We prove a general criterion for an irrational power series $f(z)=\displaystyle\sum_{n=0}^{\infty}a_nz^n$ with coefficients in a number field $K$ to admit the unit circle as a natural boundary. As an application, let $F$ be a finite field, let $d$ be a positive integer, let $A\in M_d(F[t])$ be a $d\times d$-matrix with entries in $F[t]$, and let $\zeta_A(z)$ be the Artin-Mazur zeta function associated to the multiplication-by-$A$ map on the compact abelian group $F((1/t))^d/F[t]^d$. We provide a complete characterization of when $\zeta_A(z)$ is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on $\mathbb{R}^d/\mathbb{Z}^d$ in which Baake, Lau, and Paskunas prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier together with certain patching arguments involving linear recurrence sequences.