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A multivariate, formal power series over a field $K$ is a B\'ezivin series ifall of its coefficients can be expressed as a sum of at most $r$ elements froma finitely generated subgroup $G \le K^*$; it is a P\'olya series if one cantake $r=1$. We give explicit structural descriptions of $D$-finite B\'ezivinseries and $D$-finite P\'olya series over fields of characteristic $0$, thusextending classical results of P\'olya and B\'ezivin to the multivariatesetting.

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