A general criterion for the Pólya-Carlson dichotomy and application

Abstract

We prove a general criterion for an irrational power series f ( z ) = ∑ n = 0 ∞ a n z n f(z)=\sum _{n=0}^{\infty }a_nz^n with coefficients in a number field K K to admit the unit circle as a natural boundary. As an application, let F F be a finite field, let d d be a positive integer, let A ∈ M d ( F [ t ] ) A\in M_d(F[t]) be a d × d d\times d -matrix with entries in F [ t ] F[t] , and let ζ A ( z ) \zeta _A(z) be the Artin-Mazur zeta function associated to the multiplication-by- A A map on the compact abelian group F ( ( 1 / t ) ) d / F [ t ] d F((1/t))^d/F[t]^d . We provide a complete characterization of when ζ A ( z ) \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 R d / Z d \mathbb {R}^d/\mathbb {Z}^d in which Baake, Lau, and Paskunas [Monatsh. Math. 161 (2010), pp. 33–42] 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 [Amer. Math. Soc. 373 (2020), pp. 4889–4906] together with certain patching arguments involving linear recurrence sequences.