Koike, Satoshi; Parusiński, Adam
Motivic-type invariants of blow-analytic equivalence  [ Invariants de type motivique de l'équivalence blow-analytique ]
Annales de l'institut Fourier, Tome 53 (2003) no. 7 , p. 2061-2104
Zbl 1062.14006 | MR 2044168 | 1 citation dans Numdam
doi : 10.5802/aif.2001
URL stable : http://www.numdam.org/item?id=AIF_2003__53_7_2061_0

Classification:  14B05,  32S15
Mots clés: équivalence blow-analytique, intégration motivique, fonctions zêta, formules de Thom-Sebastiani
Soit f:( d ,0)(,0) un germe de fonction analytique. On associe à f des fonctions zêta Z f,+ , Z f,- [[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:( d ,0)(,0), we associate zeta functions Z f,+ , Z f,- [[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.

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