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, solvi...