  In this paper we study the problem of deciding whether two disjoint
semialgebraic sets of an algebraic variety over R 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 the generic separation to the
separation of the two sets, yielding a decision procedure to solve the problem.
