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A subresultant theory of multivariate polynomials.

Laureano González Vega — 1990

Extracta Mathematicae

In Computer Algebra, Subresultant Theory provides a powerful method to construct algorithms solving problems for polynomials in one variable in an optimal way. So, using this method we can compute the greatest common divisor of two polynomials in one variable with integer coefficients avoiding the exponential growth of the coefficients that will appear if we use the Euclidean Algorithm. In this note, generalizing a forgotten construction appearing in [Hab], we extend the Subresultant...

A real nullstellensatz and positivstellensatz for the semipolynomials over an ordered field.

Laureano González-VegaHenri Lombardi — 1992

Extracta Mathematicae

Let K be an ordered field and R its real closure. A semipolynomial will be defined as a function from R to R obtained by composition of polynomial functions and the absolute value. Every semipolynomial can be defined as a straight-line program containing only instructions with the following type: polynomial, absolute value, sup and inf and such a program will be called a semipolynomial expression. It will be proved, using the ordinary real positivstellensatz, a general real positivstellensatz concerning...

Multivariate Sturm-Habicht sequences: real root counting on n-rectangles and triangles.

Laureano González-VegaGuadalupe Trujillo — 1997

Revista Matemática de la Universidad Complutense de Madrid

The main purpose of this note is to show how Sturm-Habicht Sequence can be generalized to the multivariate case and used to compute the number of real solutions of a polynomial system of equations with a finite number of complex solutions. Using the same techniques, some formulae counting the number of real solutions of such polynomial systems of equations inside n-dimensional rectangles or triangles in the plane are presented.

A new rational and continuous solution for Hilbert's 17th problem.

Charles N. DelzellLaureano González-VegaHenri Lombardi — 1992

Extracta Mathematicae

In this note it is presented a new rational and continuous solution for Hilbert's 17th problem, which asks if an everywhere positive polynomial can be expressed as a sum of squares of rational functions. This solution (Theorem 1) improves the results in [2] in the sense that our parametrized solution is continuous and depends in a rational way on the coefficients of the problem (what is not the case in the solution presented in [2]). Moreover our method simplifies the proof and it is easy to generalize...

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