Separation of semialgebraic sets

In this paper we study the problem of deciding whether two disjoint semialgebraic sets of an algebraic variety over are separable by a polynomial. For that we isolate a dense subfamily of Spaces of Orderings, named Geometric, which suffice to test separation and that reduce the problem to the study of the behaviour of the semialgebraic sets in their boundary. Then we derive several characterizations for the generic separation, among which there is a Geometric Criterion that can be tested algorithmically. Finally we show how to check recursively whether we can pass from generic separation to separation, obtaining a decision procedure for solving the problem.