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We present a new algorithm for solving basic parametric constructible or semi-algebraic systems like C = {x ∈ C n , p 1 (x) = 0, . . . , p s (x) = 0, f 1 (x) = 0, . . . , f l (is the set of parameters and X = [X d+1 , . . . , X n ] the set of unknowns.If Π U denotes the canonical projection on the parameter's space, solving C or S remains to compute sub-manifoldsis an analytic covering of U (we say that U has the (Π U , C)-covering property). This guarantees that the cardinal of Π −1 U ( ) ∩ C is locally constant on U and that Π −1 U (U) ∩ C is a finite collection of sheets which are all locally homeomorphic to U. In the case where Π U (C) is dense in C d , all the known algorithms for solving C or S compute implicitly or explicitly a Zariski closed subset W such that any sub-manifold of C d \ W have the (Π U , C)-covering property.We introduce the discriminant varieties of C w.r.t. Π U which are algebraic sets with the above property (even in the cases where Π U is not dense in C d ). We then show that the set of points of Π U (C) which do not have any neighborhood with the (Π U , C)-covering property is a Zariski closed set and thus the minimal discriminant variety of C w.r.t. Π U and we propose an algorithm to compute it efficiently. Thus, solving the parametric system C (resp. S) then remains to describe C d \ W D (resp. R d \ W D ) which can be done using critical points method or partial CAD based strategies.We did not fully study the complexity, but in the case of systems where Π U (C) = C d , the degree of the minimal discriminant variety as well as the running time of an algorithm able to compute it are singly exponential in the number of variables according to already known results.

Finding one point on each semi-algebraically connected component of a real algebraic variety, or at least deciding if such a variety is empty or not, is a fundamental problem of computational real algebraic geometry. Although numerous studies have been done on the subject, only a small number of efficient implementations exist.In this paper, we propose a new efficient and practical algorithm for computing such points. By studying the critical points of the restriction to the variety of the distance function to one well chosen point, we show how to provide a set of zerodimensional systems whose zeros contain at least one point on each semi-algebraically connected component of the studied variety, without any assumption either on the variety (smoothness or compactness for example) or on the system of equations which define it.From the output of our algorithm, one can then apply, for each computed zerodimensional system, any symbolic or numerical algorithm for counting or approximating the real solutions. We report some experiments using a set of pure exact methods. The practical efficiency of our method is due to the fact that we do not apply any infinitesimal deformations, unlike the existing methods based on a similar strategy.