  Given a multivariate polynomial $h(X_1,...,X_n)$ with integral coefficients
verifying an hypothesis of analytic regularity (and satisfying
$h(\textbf{0})=1$), we determine the maximal domain of meromorphy of the Euler
product $\prod_{p \ \textrm{prime}}h(p^{-s_1},...,p^{-s_n})$ and the natural
boundary is precisely described when it exists. In this way we extend a well
known result for one variable polynomials due to Estermann from 1928. As an
application, we calculate the natural boundary of the multivariate Euler
products associated to a family of toric varieties.
