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

Abstract

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 $\bar{\mathbb{Q}}$. In the special case that the coefficients are algebraic integers, our main result says that if $$F(x_1,\dots ,x_m)=\sum f(n_1,\dots ,n_m)x_1^{n_1}\cdots x_m^{n_m}$$ is a D-finite power series in $m$ variables with algebraic integer coefficients and if the logarithmic Weil height of $f(n_1,\dots ,n_m)$ is $o(n_1+\cdots +n_m)$, then $F$ is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of $F$ has the form $1-\hat{I}\P x_1^{q_1}\cdots x_m^{q_m}$ where $\hat{I}\P$ is a root of unity and $q_1,\dots ,q_m$ are nonnegative integers, not all of which are zero.