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.

[1] O.M. Abderrahmane Yacoub Polyèdre de Newton et trivialité en famille, J. Math. Soc. Japan, Volume 54 (2002), pp. 513-550 | DOI | MR | Zbl

[2] E. Bierstone; P.D. Milman Arc-analytic functions, Invent. Math., Volume 101 (1990), pp. 411-424 | DOI | MR | Zbl

[3] E. Bierstone; P.D. Milman Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant, Invent. Math., Volume 128 (1997), pp. 207-302 | DOI | MR | Zbl

[4] J. Damon; T. Gaffney Topological triviality of deformations of functions and Newton filtrations, Invent. Math., Volume 72 (1983), pp. 335-358 | DOI | MR | Zbl

[5] J. Denef; F. Loeser Motivic Igusa zeta functions, J. Alg. Geom., Volume 7 (1998), pp. 505-537 | MR | Zbl

[6] J. Denef; F. Loeser Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., Volume 135 (1999), pp. 201-232 | DOI | MR | Zbl

[7] J. Denef; F. Loeser Motivic exponential integrals and a motivic Thom-Sebastiani Theorem, Duke Math. J., Volume 99 (1999), pp. 289-309 | MR | Zbl

[8] J. Denef; F. Loeser Geometry of arc spaces of algebraic varieties, European Congress of Math. (Barcelona, July 10-14, 2000), Volume Vol. 1 (2001), pp. 327-348 | Zbl

[9] J. Denef; F. Loeser Lefschetz numbers of iterates of the monodromy and truncated arcs, Topology, Volume 41 (2002), pp. 1031-1040 | DOI | MR | Zbl

[10] T. Fukui; E. Yoshinaga The modified analytic trivialization of family of real analytic functions, Invent. Math., Volume 82 (1985), pp. 467-477 | DOI | MR | Zbl

[11] T. Fukui Seeking invariants for blow-analytic equivalence, Comp. Math., Volume 105 (1997), pp. 95-107 | DOI | MR | Zbl

[12] T. Fukui; S. Koike; T.-C. Kuo; (T. Fukuda, T. Fukui Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities (Pitman Research Notes in Math. Series), Volume 381 (1998), pp. 8-29 | Zbl

[13] T. Fukui; L. Paunescu Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan, Volume 52 (2000), pp. 433-446 | DOI | MR | Zbl

[14] J.-P. Henry; A. Parusiński Existence of Moduli for bi-Lipschitz equivalence of analytic functions, Comp. Math., Volume 136 (2003), pp. 217-235 | DOI | MR | Zbl

[15] J.-P. Henry; A. Parusiński Invariants of bi-Lipschitz equivalence of real analytic functions Banach Center Publications (to appear) | MR | Zbl

[16] H. Hironaka Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math., Volume 79 (1964), pp. 109-302 | DOI | MR | Zbl

[17] S. Izumi; S. Koike; T.-C. Kuo Computations and Stability of the Fukui Invariant, Comp. Math., Volume 130 (2002), pp. 49-73 | DOI | MR | Zbl

[18] M. Kontsevich (1995) (Lecture at Orsay, December 7)

[19] W. Kucharz Examples in the theory of sufficiency of jets, Proc. Amer. Math. Soc., Volume 96 (1986), pp. 163-166 | DOI | MR | Zbl

[20] N. Kuiper; R.D. Anderson ed. $C^1$-equivalence of functions near isolated critical points, Symp. Infinite Dimensional Topology, Baton Rouge, 1967 (Annales of Math. Studies), Volume 69 (1972), pp. 199-218 | Zbl

[21] T.-C. Kuo On $C^0$-sufficiency of jets of potential functions, Topology, Volume 8 (1969), pp. 167-171 | DOI | MR | Zbl

[22] T.-C. Kuo The modified analytic trivialization of singularities, J. Math. Soc. Japan, Volume 32 (1980), pp. 605-614 | DOI | MR | Zbl

[23] T.-C. Kuo On classification of real singularities, Invent. Math., Volume 82 (1985), pp. 257-262 | DOI | MR | Zbl

[24] K. Kurdyka Ensembles semi-algébriques symétriques par arcs, Math. Ann., Volume 282 (1988), pp. 445-462 | DOI | MR | Zbl

[25] K. Kurdyka Injective endomorphisms of real algebraic sets are surjective, Math. Ann., Volume 282 (1998), pp. 1-14 | MR | Zbl

[26] LÊ Dung Tráng Topologie des singularités des hypersurfaces complexes, Singularités à Cargèse (Astérisque), Volume 7 & 8 (1973), pp. 171-182 | Numdam | Zbl

[27] S. Lojasiewicz Ensembles semi-analytiques, I.H.E.S., 1965

[28] E. Looijenga Motivic Measures, Séminaire Bourbaki, exposé 874, mars 2000 | Numdam | Zbl

[29] C. McCrory; A. Parusiński Complex monodromy and the topology of real algebraic sets, Comp. Math., Volume 106 (1997), pp. 211-233 | DOI | MR | Zbl

[30] J. Milnor; P. Orlik Isolated singularities defined by weighted homogeneous polynomials, Topology, Volume 9 (1970), pp. 385-393 | DOI | MR | Zbl

[31] T. Nishimura Topological invariance of weights for weighted homogeneous singularities, Kodai Math. J., Volume 9 (1986), pp. 188-190 | DOI | MR | Zbl

[32] R. Quarez Espace des germes d'arcs réels et série de Poincaré d'un ensemble semi-algébrique, Ann. Inst. Fourier, Volume 51 (2001) no. 1, pp. 43-67 | DOI | Numdam | MR | Zbl

[33] O. Saeki Topological invariance of weights for weighted homogeneous isolated singularities in ${ℂ}^3$, Proc. Amer. Math. Soc., Volume 103 (1988), pp. 995-999 | MR | Zbl

[34] B. Teissier Cycles évanescents, sections planes, et conditions de Whitney, Singularités à Cargèse (Astérisque), Volume 7 & 8 (1973), pp. 285-362 | Numdam | Zbl

[35] W. Veys The topological zeta function associated to a function on a normal surface germ, Topology, Volume 38 (1999), pp. 439-456 | DOI | MR | Zbl

[36] S.-T. Yau Topological types and multiplicity of isolated quasihomogeneous surface singularities, Bull. Amer. Math. Soc., Volume 19 (1988), pp. 447-454 | DOI | MR | Zbl

[37] E. Yoshinaga; M. Suzuki Topological types of quasihomogeneous singularities in ${ℂ}^2$, Topology, Volume 18 (1979), pp. 113-116 | DOI | MR | Zbl