Solving parametric polynomial systems

We present a new algorithm for solving basic parametric constructible or semi-algebraic systems like C = {x ∈ C, p1(x) = 0, , ps(x) = 0, f1(x) 0, , fl(x) 0} or S = {x ∈ R, p1(x) = 0, , ps(x) = 0, f1(x)> 0, , fl(x)> 0}, where pi, fi ∈Q[U , X], U = [U1, , Ud] is the set of parameters and X = [Xd+1, , Xn] the set of unknowns. If ΠU denotes the canonical projection onto the parameter’s space, solving C or S is reduced to the computation of submanifolds U ⊂C (resp. U ⊂R) such that (Π U (U)∩C , ΠU) is an analytic covering of U (we say that U has the (ΠU , C)-covering property ). This guarantees that the cardinality of ΠU (u) ∩ C is constant on a neighborhood of u, that ΠU (U) ∩ C is a finite collection of sheets and that ΠU is a local diffeomorphism from each of these sheets onto U . We show that the complement in ΠU(C) (the closure of ΠU(C) for the usual topology of C) of the union of the open subsets of ΠU(C) which have the (ΠU , C)-covering property is a Zariski closed and thus is the minimal discriminant variety of C wrt. ΠU, denoted WD. We propose an algorithm to compute WD efficiently. The variety WD can then be used to solve the parametric system C (resp. S) as long as one can describe ΠU(C) \WD (resp. R d ∩ (ΠU(C) \WD)), which can be done by using critical points method or an ”open” Cylindrical Algebraic Variety.