On the practical computation of one point in each connected component of a semi-algebraic set defined by a polynomial system of equations and non-strict inequalities

Given polynomials f1, . . . , fk, g1, . . . , gs inQ[X1, . . . , Xn], we consider the semi-algebraic set S defined by: f1 = . . . = fk = 0 g1 ≥ 0, . . . , gs ≥ 0 and focus on the problem of computing at least one point in each connected component of S. We first study how to solve this problem by considering S as the union of solutions sets of polynomial systems of equations and strict inequalities and proceed to the complexity analysis of the underlying algorithm. Then, we improve this approach by proving that computing at least one point in each connected component of S can be done by computing at least one point in each connected component of real algebraic sets defined by vanishing the polynomials f1, . . . , fk and some of the polynomials g1, . . . , gs. The complexity analysis shows that this latter approach is better than the former one. Finally, we present our implementation and use it to solve an application in Pattern Matching.