Motivic-type invariants of blow-analytic equivalence

Koike, Satoshi; Parusiński, Adam

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

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

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

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