D-finite multivariate series with arithmetic restrictions on their coefficients

Abstract

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 $G \le K^*$ ; 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.