Gamboa Mutuberria, José ManuelFernando Galván, José Francisco2023-06-202023-06-2020120129-167X10.1142/S0129167X12500310https://hdl.handle.net/20.500.14352/42179In 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, we give full characterizations of the families of semialgebraic sets for which these classical results are true.journal articlehttp://www.worldscinet.com/ijm/ijm.shtmlhttp://www.worldscinet.commetadata only access512.7Nash functionNash setirreducible semialgebraic setirreducible components of a semialgebraic setw-Nash setq-Nash setsubstitution theorempositivstellens ¨atze17th Hilbert problem and real nullstellensatzGeometria algebraica1201.01 Geometría Algebraica