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 ; Parusiński, Adam

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 2044168 | Zbl 1062.14006

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

@article{AIF_2003__53_7_2061_0,
     author = {Koike, Satoshi and Parusi\'nski, Adam},
     title = {Motivic-type invariants of blow-analytic equivalence},
     journal = {Annales de l'Institut Fourier},
     pages = {2061--2104},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     doi = {10.5802/aif.2001},
     mrnumber = {2044168},
     zbl = {1062.14006},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2001/}
}

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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