2022/06/02 by Jason P. Bell, Keira Gunn, Bell, Jason P. +5 · 1 citation
Computer Science · Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2206.00862
openalex publication_date 2022/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a general criterion for an irrational power series f(z)=∑n=0∞anzn 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∈ Md(F[t]) be a d× d-matrix with entries in F[t], and let ζA(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 ζA(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 ℝd/ℤd 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.