ON THE IRREDUCIBLE COMPONENTS OF A SEMIALGEBRAIC SET

In this work we define a semialgebraic set S C R-n to be irreducible if the noetherian ring N(S) of Nash functions on S is an integral domain. Keeping this notion we develop a satisfactory theory of irreducible components of semialgebraic sets, and we use it fruitfully to approach four classical problems in Real Geometry for the ring N(S): Substitution Theorem, Positivstellensatze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, we give full characterizations of the families of semialgebraic sets for which these classical results are true.