Motivic-type invariants of blow-analytic equivalence

Koike, Satoshi; Parusiński, Adam

doi:10.5802/aif.2001

Koike, Satoshi; Parusiński, Adam

Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2061-2104.

Abstract
Résumé

To a given analytic function germ $f : (ℝ^{d}, 0) \to (ℝ, 0)$ , we associate zeta functions $Z_{f, +}$ , $Z_{f, -} \in ℤ [[T]]$ , 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.

Soit $f : (ℝ^{d}, 0) \to (ℝ, 0)$ un germe de fonction analytique. On associe à $f$ des fonctions zêta $Z_{f, +}$ , $Z_{f, -} \in ℤ [[T]]$ 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.

MR: 2044168 | Zbl: 1062.14006

DOI: 10.5802/aif.2001
Classification: 14B05, 32S15
Keywords: blow-analytic equivalence, motivic integration, zeta functions, Thom-Sebastiani formulae

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Koike, Satoshi; Parusiński, Adam. Motivic-type invariants of blow-analytic equivalence. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2061-2104. doi : 10.5802/aif.2001. https://aif.centre-mersenne.org/articles/10.5802/aif.2001/

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