How to count zeroes of polynomials on quadrature domains using the Bezout matrix

TL;DR

This paper extends the theory of Bezoutians to real Riemann surfaces, linking their signatures to topological data, and introduces a method to count polynomial zeros in quadrature domains via Bezoutian inertia.

Contribution

It generalizes Bezoutian theory to real Riemann surfaces and proposes a novel approach to count polynomial zeros in quadrature domains using Bezoutian inertia.

Findings

Signature of Bezoutian relates to topological data of quotient functions.

Method effectively counts zeros of polynomials in quadrature domains.

Examples demonstrate application in simply connected quadrature domains.

Abstract

Classically, the Bezout matrix or simply Bezoutian of two polynomials is used to locate the roots of the polynomial and, in particular, test for stability. In this paper, we develop the theory of Bezoutians on real Riemann surfaces of dividing type. The main result connects the signature of the Bezoutian of two real meromorphic functions to the topological data of their quotient, which can be seen as the generalization of the classical Cauchy index. As an application, we propose a method to count the number of zeroes of a polynomial in a quadrature domain using the inertia of the Bezoutian. We provide examples of our method in the case of simply connected quadrature domains.