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Nombres de Betti virtuels des ensembles symétriques par arcs et équivalence de Nash après éclatements

Abstract : Motivic integration, a theory recently developped by J. Denef and F. Loeser, is a powerful tool for constructing invariants of singularities. However it necessitates, in order to construct measures for this integration, to known generalized Euler characteristics, that is additive and multiplicative invariants at the level of varieties. In the setting of real algebraic geometry, such generalized Euler characteristics does not abound whereas it is the case in complex algebraic geometry. We construct in this Ph. D. thesis such an invariant, called the virtual Poincaré polynomial, for the larger category or arc-symmetric sets, generalizing a result of C. McCrory and A. Parusiński. We prove that this virtual Poincaré polynomial is moreover an invariant for Nash isomorphisms between arc-symmetric sets. This enables us, following the work of J. Denef and F. Loeser, to construct zeta functions for a Nash function germ. In particular, we can state a formula for these zeta functions in terms of a modification of the germ. The main result about these zeta functions is that they are invariants for the blow-Nash equivalence between Nash function germs, which is a particular case of the blow-analytic equivalence due to T.-C. Kuo. Moreover, we prove a triviality result concerning the blow-Nash equivalence, whose proof requires new techiques because of the particularity of the Nash setting.
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Contributor : Goulwen Fichou <>
Submitted on : Thursday, January 22, 2004 - 2:27:12 PM
Last modification on : Tuesday, May 7, 2019 - 6:30:09 PM
Long-term archiving on: : Tuesday, September 7, 2010 - 4:52:46 PM


  • HAL Id : tel-00004279, version 1


Goulwen Fichou. Nombres de Betti virtuels des ensembles symétriques par arcs et équivalence de Nash après éclatements. Mathématiques [math]. Université d'Angers, 2003. Français. ⟨tel-00004279⟩



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