D-finiteness, rationality, and height III: multivariate Pólya–Carlson dichotomy

We prove a result that can be seen as an analogue of the Pólya–Carlson theorem for multivariate D-finite power series with coefficients in \(\bar{\mathbb {Q}}\) $\overline{\mathbb{Q}}$ . In the special case that the coefficients are algebraic integers, our main result says that if \(\begin{aligned} F(x_1,\ldots ,x_m)=\sum f(n_1,\ldots ,n_m)x_1^{n_1}\cdots x_m^{n_m} \end{aligned}\) $$\begin{array}{c}\hfill F(x_1,\dots ,x_m)=\sum f(n_1,\dots ,n_m)x_1^{n_1}\cdots x_m^{n_m}\end{array}$$ is a D-finite power series in m variables with algebraic integer coefficients and if the logarithmic Weil height of \(f(n_1,\ldots ,n_m)\) $f(n_1,\dots ,n_m)$ is \(o(n_1+\cdots +n_m)\) $o(n_1+\cdots +n_m)$ , then F is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of F has the form \(1-\zeta x_1^{q_1}\cdots x_m^{q_m}\) $1-\zeta x_1^{q_1}\cdots x_m^{q_m}$ where \(\zeta \) $\zeta$ is a root of unity and \(q_1,\ldots ,q_m\) $q_1,\dots ,q_m$ are nonnegative integers, not all of which are zero.