On the solution set of a sparse polynomial equation system

GABRIELA JERONIMO

Conferencia; I Latin American School of Algebraic Geometry and Applications; 2011

We will focus on recent symbolic algorithms to solve systems of sparse multivariatepolynomial equations, namely, systems of equations given by polynomials with nonzerocoefficients only at prescribed sets of monomials.The structure of the prescribed monomial sets, the so-called family of supports ofthe system, is closely related to geometric properties of the set of its complex solutions.A fundamental result in sparse elimination is Bernstein´s theorem (1975), which statesthat the number of isolated roots in (C*)^n of a generic system of n equations inn variables with given supports is the mixed volume of the family of supports.In this talk we will present a symbolic probabilistic algorithm to compute all theisolated roots in C^n of an arbitrary sparse polynomial system and an upper boundfor the number of these roots. In addition, we will show combinatorial conditions onthe system supports that enable us to describe algorithmically the equidimensionalcomponents of positive dimension of generic sparse polynomial systems within goodcomplexity bounds.This is joint work with María Isabel Herrero and Juan Sabia.