$D$-finite multivariate series with arithmetic restrictions on their coefficients

Abstract

A multivariate, formal power series over a field $K$ is a Bézivin series if all of its coefficients can be expressed as a sum of at most $r$ elements from a finitely generated subgroup $G\le K^{\ast }$; it is a Pólya series if one can take $r=1$. We give explicit structural descriptions of $D$-finite Bézivin series and $D$-finite Pólya series over fields of characteristic $0$, thus extending classical results of Pólya and Bézivin to the multivariate setting.