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 Q¯. In the special case that the coefficients are algebraic integers, our main result says that if (Formula presented.) is a D-finite power series in m variables with algebraic integer coefficients and if the logarithmic Weil height of f(n1,…,nm) is o(n1+⋯+nm), then F is a rational function and, up to scalar multiplication, every irreducible factor of the denominator of F has the form 1-ζx1q1⋯xmqm where ζ is a root of unity and q1,…,qm are nonnegative integers, not all of which are zero.