Holomorphic Rational Functions of Several Variables and Sums of Squares of Polynomials

TL;DR

This paper establishes conditions under which the numerator of the partial derivative of a holomorphic rational function in several variables can be expressed as a sum of squares of polynomials, advancing understanding of such functions.

Contribution

It provides necessary and sufficient conditions for representing derivatives of holomorphic rational functions as sums of squares, a novel result in multivariable complex analysis.

Findings

Identifies conditions for sum of squares representation

Advances theory of holomorphic rational functions in multiple variables

Connects derivatives to polynomial sum of squares

Abstract

Necessary and sufficient conditions are obtained under which the numerator of the partial derivative of a rational function holomorphic in open upper poly-halfplane is the sum of squares of polynomials.