Motivic-type invariants of blow-analytic equivalence

[Invariants de type motivique de l'équivalence blow-analytique]

Koike, Satoshi ; Parusiński, Adam

Annales de l'Institut Fourier, Tome 53 (2003), p. 2061-2104 / Harvested from Numdam

Résumé

Soit $f:({\mathbb {R}}^d,0) \rightarrow ({\mathbb {R}},0)$ un germe de fonction analytique. On associe à $f$ des fonctions zêta $Z_{f,+}$, $Z_{f,-} \in {\mathbb {Z}} [[T]]$ définies de manière similaire aux fonctions zêta motiviques de Denef et Loeser. On montre que ces fonctions sont rationnelles et ne dépendent que de la classe d’équivalence blow-analytique au sens de Kuo de $f$. En utilisant ces fonctions zêta et l’invariant de Fukui on donne une classification des polynômes de Brieskorn de deux variables à équivalence blow-analytique près. Pour les polynômes de Brieskorn de trois variables on obtient une classification presque complète.

Abstract

To a given analytic function germ $f:({\mathbb {R}}^d,0) \rightarrow ({\mathbb {R}},0)$, we associate zeta functions $Z_{f,+}$, $Z_{f,-} \in {\mathbb {Z}} [[T]]$, defined analogously to the motivic zeta functions of Denef and Loeser. We show that our zeta functions are rational and that they are invariants of the blow-analytic equivalence in the sense of Kuo. Then we use them together with the Fukui invariant to classify the blow-analytic equivalence classes of Brieskorn polynomials of two variables. Except special series of singularities our method classifies as well the blow-analytic equivalence classes of Brieskorn polynomials of three variables.

Détail

Publié le : 2003-01-01

MR 2044168 | Zbl 1062.14006 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2001
Classification:  14B05,  32S15
Mots clés: équivalence blow-analytique, intégration motivique, fonctions zêta, formules de Thom-Sebastiani

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