Residue Integrals and Waring's Formulas for a Class of Systems of Transcendental Equations in $\mathbb{C}^n$

TL;DR

This paper develops a method to compute residue integrals for certain transcendental systems in complex space, enabling the derivation of Waring's formulas and applications to multi-variable series summation.

Contribution

It introduces a novel approach for residue integral computation applicable to transcendental systems where algebraic methods fail.

Findings

Residue integrals can be computed over specific cycles for these systems.

Conditions are identified under which residue integrals match power sums of roots.

Application demonstrated in summing multi-variable number series.

Abstract

The present article is focused on the study of a special class of systems of nonlinear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of the study of such systems, we develop a method for computing residue integrals with integration over certain cycles. We describe conditions under which the mentioned residue integrals coincide with power sums of the inverses to the roots of a system of equations (i.e., multidimensional Waring's formulas). As an application of the suggested method, we consider a problem of finding sums of multi-variable number series.