Residue integrals and Waring’s formulas for algebraic or even transcendental systems

ABSTRACT

The present article is focused on the study of a special class of systems of non-linear transcendental equations for which classical algebraic and symbolic methods are inapplicable. For the purpose of 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). Based on this, we develop an algorithm that computes such power sums without computing the roots. As an application of the suggested method, we consider a problem of finding sums of multi-variable number series.

KEYWORDS:

• Systems of transcendental equations
• residue integrals
• power sums
• Waring’s formulas

AMS SUBJECT CLASSIFICATIONS:

• 65H10
• 68W30

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by grants of the President of the Russian Federation for young scientists [project number MD-197.2017.1] and for leading scientific schools [project number NSh-9149.2016.1], by grant of the Government of the Russian Federation for investigations under the guidance of the leading scientists of the Siberian Federal University [contract number 14.Y26.31.0006], and by the Russian Foundation for Basic Research [project number 18-01-00024-a].