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

Consider a parametric system of <i>n</i> polynomial equations and <i>r</i> polynomial inequations in <i>n</i> unknowns and <i>s</i> 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 <i>discriminant variety</i>. Let <i>d</i> 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 <i>D := (n+r)d<sup>(n+1)</sup></i>. Moreover we provide in this case a deterministic method which computes the minimal discriminant variety in σ<sup><i>O</i>(1)</sup><i>D<sup>O(n+s)</sup></i> bit-operations on a deterministic Turing machine.

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