On the solution set of a sparse polynomial equation system

JERONIMO, GABRIELA; HERRERO, MARÍA ISABEL; SABIA, JUAN

Buenos Aires - La Cumbre

Congreso; Research School on Algebraic Geometry and Applications - I Latin American School of Algebraic Geometry and Applications; 2011

CIMPA - ICTP - UNIVERSIDAD DE BUENOS AIRES

We will focus on recent symbolic algorithms to solve systems of sparse multivariate polynomial equations, namely, systems of equations given by polynomials with nonzero coefficients only at prescribed sets of monomials. The structure of the prescribed monomial sets, the so-called family of supports of the 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 states that the number of isolated roots in the torus of a generic system of n equations in n 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 the isolated roots in the affine space Cn of an arbitrary sparse polynomial system and an upper bound for the number of these roots. In addition, we will show combinatorial conditions on the system supports that enable us to describe algorithmically the equidimensional components of positive dimension of generic sparse polynomial systems within good complexity bounds. Expuesto por Gabriela Jeronimo