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Mathematics > Complex Variables

arXiv:1805.03628 (math)
[Submitted on 9 May 2018 (v1), last revised 9 May 2019 (this version, v2)]

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

Authors:Eli Shamovich, Victor Vinnikov
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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.
Comments: 23 pages, 2 figures. Comments are welcome
Subjects: Complex Variables (math.CV)
Cite as: arXiv:1805.03628 [math.CV]
  (or arXiv:1805.03628v2 [math.CV] for this version)
  https://doi.org/10.48550/arXiv.1805.03628

Submission history

From: Eli Shamovich [view email]
[v1] Wed, 9 May 2018 17:37:32 UTC (48 KB)
[v2] Thu, 9 May 2019 19:03:13 UTC (49 KB)
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