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

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
Pages (from-to)1-26
Number of pages26
JournalAdvances in Mathematics
Volume352
DOIs
StatePublished - 20 Aug 2019

Keywords

  • Bezoutian of meromorphic functions
  • Locating zeroes
  • Polynomials
  • Quadrature domains
  • Real Riemann surface

ASJC Scopus subject areas

  • Mathematics (all)

Fingerprint

Dive into the research topics of 'How to count zeroes of polynomials on quadrature domains using the Bezout matrix'. Together they form a unique fingerprint.

Cite this