The localization of solutions to systems of algebraic equations and inequalities: the Hermite method
type: Item
instance of: scholarly article
title: The localization of solutions to systems of algebraic equations and inequalities: the Hermite method
zbMATH Open document ID: 0902.12002
published in: Doklady Mathematics
publication date: 1998-03-11T00:00:00Z
review text: The authors present an algorithm for the localization of solutions of systems of algebraic equations and inequalities. Their method is based on results of É. Bézout, C. Hermite, J. J. Sylvester, L. Kronecker and A. Markoff. If \(F\) and \(G\) are univariate polynomials with real coefficients, then \(\text{nrs} \{F=0\mid G>0\}\) denotes the number of distinct real solutions of the \(F(x)=0\) that satisfy \(G(x)>0\), and a similar notation is used for bivariate polynomials. In the main result of this paper \(\text{nrs} \{F_1=F_2=0\}\) and \(\text{nrs} \{F_1=F_2=0\mid G>0\}\) are studied, where \(F_1,F_2\) and \(G\) are polynomials in two variables and the coefficients of \(F_1\) and \(F_2\) satisfy some subresultant conditions. On the other hand these techniques are used for giving sufficient and necessary conditions for the compatibility of the system of inequalities \(G_1>0\), \(G_2>0\), \(G_3>0\), where \(G_i\) are bivariate polynomials. Note that \textit{A. Yu. Uteshev} and \textit{S. G. Shulyak} [Linear Algebra Appl. 177, 49-88 (1993; Zbl 0769.65023)] studied this system for univariate polynomials.
reviewed by: Doru Ştefănescu
Mathematics Subject Classification ID: 12D05
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 13P10
zbMATH DE Number: 1113103
zbMATH Keywords: Hermite's method
zbMATH Keywords: algorithm
zbMATH Keywords: localization of solutions of systems of algebraic equations
zbMATH Keywords: univariate polynomials
zbMATH Keywords: bivariate polynomials
author: Alexei Yu. Uteshev
author: Timofei M. Cherkasov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1113103