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Ruiz Sancho, Jesús María and Shiota, Masahiro
(1994)
*On global Nash functions.*
Annales Scientifiques de l'École Normale Supérieure. Quatrième Série, 27
(1).
pp. 103-124.
ISSN 0012-9593

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Official URL: http://archive.numdam.org/ARCHIVE/ASENS/ASENS_1994_4_27_1/ASENS_1994_4_27_1_103_0/ASENS_1994_4_27_1_103_0.pdf

## Abstract

Let M superset-of R be a compact Nash manifold, and N (M) [resp. O(M)] its ring of global Nash (resp. analytic) functions. A global Nash (resp. analytic) set is the zero set of finitely many global Nash (resp. analytic) functions, and we have the usual notion of irreducible set. Then we say that separation holds for M if every Nash irreducible set is analytically irreducible. The main result of this paper is that separation holds if and only if every semialgebraic subset of M described by s global analytic inequalities can also be described by s global Nash inequalities. In passing, we also prove that when separation holds, every Nash function on a Nash set extends to a global Nash function on M.

Item Type: | Article |
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Uncontrolled Keywords: | Extension theorem; rings; separation problem; problem of equal complexities; Nash functions; number of inequalities; fans |

Subjects: | Sciences > Mathematics > Algebraic geometry Sciences > Mathematics > Set theory |

ID Code: | 19927 |

Deposited On: | 12 Feb 2013 17:12 |

Last Modified: | 12 Dec 2018 15:13 |

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