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

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.