The convergence condition for improper short integrals in terms of Newton polytopes

Abstract

The article considers multidimensional improper integrals of functions that are the product of generalized polynomials in some degrees. Such integrals are found in many branches of mathematics and theoretical physics. In particular, they include Feynman integrals arising in the study of various objects of quantum field theory. The exact calculation of these integrals is a difficult and not always possible task; therefore, determining the conditions for their convergence and obtaining their asymptotic expansion in one of the parameters is of considerable practical interest. The convergence conditions for the integrals considered in the article can still be used, for example, in the study of multiple series representing the sum of the values of a rational function at the nodes of an integer lattice. The article considers the problem when the integration area is R+^n, and the generalized polynomials included in the integrand are either positive everywhere except zero or have positive coefficients. The convergence set of these integrals is described and the equivalence of the convergence condition to the condition on the Newton polytopes of polynomials in integrands is proved. The convergence criterion proved in the paper coincides in formulation with the corresponding result of the work of A. K. Tsikh and T. O. Ermolaeva, but it was obtained by other methods and for a slightly wider set of integrands. The proofs of the statements in the paper are based on the simplest properties of convex polytopes and basic facts from the theory of improper multiple integrals.