For an algebraic number \(\alpha\), we define its denominator, denoted \(\mathrm{den}(\alpha)\), to be the smallest positive integer \(d\) such that \(d\alpha\) is an algebraic integer. Let \(S\) be a subset of \(\mathbb{N}\) such that \(\big|S\cap [1,n]\big|=o(n/\log n) \) as \(n\to \infty\). Let \(K\) be a number field and let \(f(z)=\sum a_nz^n \in K[[z]]\) such that \(\sigma(f)=\sum \sigma(a_n)z^n \) converges in the open unit disk for every embedding \(\sigma: K\to \mathbb{C}\). Suppose that for every \(\beta > 1\), we have \[ \mathrm{lcm} \lbrace \mathrm{den}(a_k) : k \leq n, k \notin S\rbrace < \beta^n\] for every sufficiently large integer \(n\).
The main result of the paper shows that either \(f(z)\) admits the unit circle as a natural boundary or there exists \(\sum b_nz^n \in K[[z]]\) that is the power series of a rational function whose poles are located at the roots of unity such that \(a_n = b_n\) for every \(n \in \mathbb{N} \backslash S\).
The above result is a generalization of the well-known Polya-Carlson dichotomy when \(K = \mathbb{Q}\) and \(S = \emptyset\) [F. Carlson, Math. Z. 11, 1–23 (1921; JFM 48.0387.01)].
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 \(F((1/t))^d/F[t]^d\). The authors 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.