Affine solution sets of sparse polynomial systems

Authors:
MaríA Isabel Herrero;Gabriela Jeronimo;Juan Sabia
Affiliations:
Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina;Departamento de Matemática and IMAS, UBA-CONICET, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, (1428) Buenos Aires, Argentina;Departamento de Ciencias Exactas, Ciclo Básico Común, Universidad de Buenos Aires, and IMAS, UBA-CONICET, Ciudad Universitaria, (1428) Buenos Aires, Argentina
Venue:
Journal of Symbolic Computation
Year:
2013

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Abstract

This paper focuses on the equidimensional decomposition of affine varieties defined by sparse polynomial systems. For generic systems with fixed supports, we give combinatorial conditions for the existence of positive dimensional components which characterize the equidimensional decomposition of the associated affine variety. This result is applied to design an equidimensional decomposition algorithm for generic sparse systems. For arbitrary sparse systems of n polynomials in n variables with fixed supports, we obtain an upper bound for the degree of the affine variety defined and we present an algorithm which computes finite sets of points representing its equidimensional components.