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.