Complexity of the resolution of parametric systems of polynomial equations and inequations

Abstract

Consider a parametric system of n polynomial equations and r polynomial inequations in n unknowns and s parameters, with rational coefficients. A recurrent problem is to determine some open set in the parameter space where the considered parametric system admits a constant number of real solutions. Following the works of Lazard and Rouillier, this can be done by the computation of a discriminant variety. Let d bound the degree of the input's polynomials, and σ bound the bit-size of their coefficients. Based on some usual assumptions for the applications we prove that the degree of the computed minimal discriminant variety is bounded by D := (n+r)d(n+1). Moreover we provide in this case a deterministic method which computes the minimal discriminant variety in σO(1)DO(n+s) bit-operations on a deterministic Turing machine.