The localization of solutions to systems of algebraic equations and inequalities: the Hermite method
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.
localization of solutions of systems of algebraic equations
scientific article; zbMATH DE number 1113103