%0 Journal Article
%A Gamboa Mutuberria, José Manuel
%A Fernando Galván, José Francisco
%T On The Irreducible Components Of A Semialgebraic Set
%D 2012
%@ 0129-167X
%U https://hdl.handle.net/20.500.14352/42179
%X In this work we define a semialgebraic set S Rn to be irreducible if the noetherian ring 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 : Substitution Theorem, Positivstellens¨atze, 17th Hilbert Problem and real Nullstellensatz, whose solution was known just in case S = M is an affine Nash manifold. In fact, wegive full characterizations of the families of semialgebraic sets for which these classical results are true.
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