On the integral criteria for a convergence of multidimensional Dirichlet series

We consider Dirichlet series associated with a set of $m$ polynomials in $n$ variables. Such series depend on $m$ complex parameters. They were studed by B. Lichtin and others in the case of hypoelliptic polynomials. We consider a more general class of polynomials so called quasielliptic polynomials in the sence of T. Ermolaeva and A. Tsikh. Using the toric geometry we discribe the domain of convergence in terms of Newton polytopes of polynomials defining the series. As an auxiliary statement we give a criterion for convergence of some integrals over $\mathbb {R}^n$.