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

TL;DR

This paper characterizes multivariate $D$-finite series with coefficients constrained by arithmetic restrictions, extending classical univariate results to a multivariate context.

Contribution

It provides explicit structural descriptions of $D$-finite Bézivin and Pólya series in multivariate settings over characteristic zero fields.

Findings

Structural descriptions of $D$-finite Bézivin series

Structural descriptions of $D$-finite Pólya series

Extension of classical univariate results to multivariate case

Abstract

A multivariate, formal power series over a field $K$ is a B\'ezivin 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 P\'olya series if one can take $r=1$. We give explicit structural descriptions of $D$-finite B\'ezivin series and $D$-finite P\'olya series over fields of characteristic $0$, thus extending classical results of P\'olya and B\'ezivin to the multivariate setting.