Solving parametric polynomial systems

We present a new algorithm for solving basic parametric constructible or semi-algebraic systems of the form C={x∈Cn,p1(x)=0,…,ps(x)=0,f1(x)≠0,…,fl(x)≠0} or S={x∈Rn,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⊂Cd or U⊂Rd such that is an analytic covering of U (we say that U has the (ΠU,C)-covering property). This guarantees that the cardinality of is constant on a neighborhood of u, that 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 (the closure of ΠU(C) for the usual topology of Cn) of the union of all the open subsets of which have the (ΠU,C)-covering property is a Zariski closed set which is called the minimal discriminant variety of C w.r.t. ΠU, denoted as 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 (resp. ). This can be done by using the critical points method or an “open” cylindrical algebraic decomposition.