The Hermitian Killing form and root counting of complex polynomials with conjugate variables

TL;DR

This paper introduces a Hermitian form to analyze solutions of complex polynomial systems with conjugate variables, leading to a new bound on the number of solutions of harmonic polynomial equations.

Contribution

It presents a novel Hermitian form that encodes solution information and establishes a new general bound for solutions of harmonic polynomial equations.

Findings

Introduces a Hermitian form for complex polynomial systems

Provides a new bound on the number of solutions of harmonic polynomials

Extends classical solution counting methods to complex conjugate variables

Abstract

Inspired by the work about solutions of a system of real polynomial equations done by Hermite, this paper introduces a Hermitian form, which encodes information about solutions of a system of complex polynomial equations with conjugate variables. Adopting the presented object, a new general bound for the number of solutions of an harmonic polynomial equation is proved.