  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.
