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The localization of solutions to systems of algebraic equations and inequalities: the Hermite method. (English. Russian original) Zbl 0902.12002
Dokl. Math. 53, No. 2, 227-229 (1996); translation from Dokl. Akad. Nauk 347, No. 4, 451-453 (1996).
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 A. Yu. Uteshev and S. G. Shulyak [Linear Algebra Appl. 177, 49-88 (1993; Zbl 0769.65023)] studied this system for univariate polynomials.

##### MSC:
 12D05 Polynomials in real and complex fields: factorization 12Y05 Computational aspects of field theory and polynomials (MSC2010) 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)