### Impacto

Gamboa Mutuberria, José Manuel and Fernando Galván, José Francisco
(2012)
*On The Irreducible Components Of A Semialgebraic Set.*
International Journal Of Mathematics, 23
(4).
p. 40.
ISSN 0129-167X

Official URL: http://www.worldscinet.com/ijm/ijm.shtml

## Abstract

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, we

give full characterizations of the families of semialgebraic sets for which these classical results are true.

Item Type: | Article |
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Uncontrolled Keywords: | Nash function; Nash set; irreducible semialgebraic set; irreducible components of a semialgebraic set; w-Nash set; q-Nash set; substitution theorem; positivstellens ¨atze; 17th Hilbert problem and real nullstellensatz |

Subjects: | Sciences > Mathematics > Algebraic geometry |

ID Code: | 15390 |

Deposited On: | 28 May 2012 09:02 |

Last Modified: | 28 May 2012 09:02 |

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