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Etude quantitative des ensembles semi-pfaffiens

Abstract : In the present thesis, we establish upper-bounds on the Betti numbers of sets defined using Pfaffian functions, in terms of the natural Pfaffian complexity (or format) of those sets. Pfaffian functions were introduced by Khovanskii, as solutions of certain polynomial differential systems that have polynomial-like behaviour over the real domain. Semi-Pfaffian sets are sets that satisfy a quantifier-free sign condition on such functions, and sub-Pfaffian sets are linear projection of semi-Pfaffian sets. Wilkie showed that Pfaffian functions generate an o-minimal structure, and Gabrielov showed that this structure could be effectively described by Pfaffian limit sets. Using Morse theory, deformations, recursion on combinatorial levels and a spectral sequence associated to continuous surjections, we give in this thesis effective estimates for sets belonging to all of the above classes.
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Contributor : Thierry Zell <>
Submitted on : Monday, February 14, 2005 - 6:41:40 PM
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  • HAL Id : tel-00008488, version 1


Thierry Zell. Etude quantitative des ensembles semi-pfaffiens. Mathématiques [math]. Université Rennes 1, 2003. Français. ⟨tel-00008488⟩



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