Motivic-type invariants of blow-analytic equivalence

Koike, Satoshi; Parusiński, Adam

doi:10.5802/aif.2001

Motivic-type invariants of blow-analytic equivalence
[Invariants de type motivique de l'équivalence blow-analytique]

Koike, Satoshi ¹ ; Parusiński, Adam ²

¹ Hyogo University of Teacher Education, Department of Mathematics, 942-1 Shimokume, Kato, Yashiro, Hyogo 673-1494 (Japon)
² Université d'Angers, Département de Mathématiques, 2 Bd Lavoisier, 49045 Angers Cedex (France)

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

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/aif.2001

Classification : 14B05, 32S15
Keywords: blow-analytic equivalence, motivic integration, zeta functions, Thom-Sebastiani formulae
Mot clés : équivalence blow-analytique, intégration motivique, fonctions zêta, formules de Thom-Sebastiani

Affiliations des auteurs :

Koike, Satoshi ¹ ; Parusiński, Adam ²

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     title = {Motivic-type invariants of blow-analytic equivalence},
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     pages = {2061--2104},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
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Koike, Satoshi; Parusiński, Adam. Motivic-type invariants of blow-analytic equivalence. Annales de l'Institut Fourier, Tome 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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