Extension of Estermann's theorem to Euler products associated to a multivariate polynomial

Given a multivariate polynomial $h(X_1,\mathrm{.}\mathrm{.}\mathrm{.},X_n)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h(\mathbf{\text{0}})=1$), we determine the maximal domain of meromorphy of the Euler product ${\prod }_{p\text{ prime}}h(p^{-s_1},\mathrm{.}\mathrm{.}\mathrm{.},p^{-s_n})$ 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.