Basicness of Semialgebraic Sets

This paper is concerned with the problem of deciding whether a semialgebraic set S of an algebraic variety X over R is basic. Furthermore, in such a case, we decide what is the sharp number of inequalities defining S. For that, it suffices to desingularize X, as well as the boundary of S, and then ask the same question for the trace of S on its boundary. In this way, after a finite number of blowing-ups, we lower the dimension of the data and by induction we get a finite decision procedure to solve this problem. Decidability of other known criteria is also analyzed.