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Satoshi Koike; Adam Parusiński
Motivic-type invariants of blow-analytic equivalence
(Invariants de type motivique de l'équivalence blow-analytique)
Annales de l'institut Fourier, 53 no. 7 (2003), p. 2061-2104, doi: 10.5802/aif.2001
Article PDF | Reviews MR 2044168 | Zbl 1062.14006 | 1 citation in Cedram
Class. Math.: 14B05, 32S15
Keywords: blow-analytic equivalence, motivic integration, zeta functions, Thom-Sebastiani formulae
To a given analytic function germ $f:({\Bbb R}^d,0) \to ({\Bbb R},0)$, we associate zeta
functions $Z_{f,+}$, $Z_{f,-} \in {\Bbb Z} [[T]]$, 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.
[1] O.M. Abderrahmane Yacoub, “Polyèdre de Newton et trivialité en famille”, J. Math. Soc. Japan 54 (2002), p. 513-550 Article | MR 1900955 | Zbl 1031.58024
[2] E. Bierstone & P.D. Milman, “Arc-analytic functions”, Invent. Math. 101 (1990), p. 411-424 Article | MR 1062969 | Zbl 0723.32005
[3] E. Bierstone & P.D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math. 128 (1997), p. 207-302 Article | MR 1440306 | Zbl 0896.14006
[4] J. Damon & T. Gaffney, “Topological triviality of deformations of functions and Newton filtrations”, Invent. Math. 72 (1983), p. 335-358 Article | MR 704395 | Zbl 0519.58021
[5] J. Denef & F. Loeser, “Motivic Igusa zeta functions”, J. Alg. Geom. 7 (1998), p. 505-537 MR 1618144 | Zbl 0943.14010
[6] J. Denef & F. Loeser, “Germs of arcs on singular algebraic varieties and motivic integration”, Invent. Math. 135 (1999), p. 201-232 Article | MR 1664700 | Zbl 0928.14004
[7] J. Denef & F. Loeser, “Motivic exponential integrals and a motivic Thom-Sebastiani Theorem”, Duke Math. J. 99 (1999), p. 289-309 Article | MR 1708026 | Zbl 0966.14015
[8] J. Denef & F. Loeser, Geometry of arc spaces of algebraic varieties, 2001, p. 327-348 Zbl 01944722
[9] J. Denef & F. Loeser, “Lefschetz numbers of iterates of the monodromy and truncated arcs”, Topology 41 (2002), p. 1031-1040 Article | MR 1923998 | Zbl 1054.14003
[10] T. Fukui & E. Yoshinaga, “The modified analytic trivialization of family of real analytic functions”, Invent. Math. 82 (1985), p. 467-477 Article | MR 811547 | Zbl 0559.58005
[11] T. Fukui, “Seeking invariants for blow-analytic equivalence”, Comp. Math. 105 (1997), p. 95-107 Article | MR 1436747 | Zbl 0873.32008
[12] T. Fukui, S. Koike & T.-C. Kuo, Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities, Pitman Research Notes in Math. Series, 1998, p. 8-29 Zbl 0954.26012
[13] T. Fukui & L. Paunescu, “Modified analytic trivialization for weighted homogeneous function-germs”, J. Math. Soc. Japan 52 (2000), p. 433-446 Article | MR 1742795 | Zbl 0964.32023
[14] J.-P. Henry & A. Parusiński, “Existence of Moduli for bi-Lipschitz equivalence of analytic functions”, Comp. Math. 136 (2003), p. 217-235 Article | MR 1967391 | Zbl 1026.32055
[15] J.-P. Henry & A. Parusiński, “Invariants of bi-Lipschitz equivalence of real analytic functions”, Banach Center Publications (to appear) MR 2104338 | Zbl 1059.32006
[16] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II”, Ann. of Math. 79 (1964), p. 109-302 Article | MR 199184 | Zbl 0122.38603
[17] S. Izumi, S. Koike & T.-C. Kuo, “Computations and Stability of the Fukui Invariant”, Comp. Math. 130 (2002), p. 49-73 Article | MR 1883691 | Zbl 1007.58023
[18] M. Kontsevich, “”, Lecture at Orsay, December 7, 1995
[19] W. Kucharz, “Examples in the theory of sufficiency of jets”, Proc. Amer. Math. Soc. 96 (1986), p. 163-166 Article | MR 813830 | Zbl 0594.58008
[20] N. Kuiper, $C^1$-equivalence of functions near isolated critical points, Annales of Math. Studies, Princeton Univ. Press, 1972, p. 199-218 Zbl 0236.58001
[21] T.-C. Kuo, “On $C^0$-sufficiency of jets of potential functions”, Topology 8 (1969), p. 167-171 Article | MR 238338 | Zbl 0183.04601
[22] T.-C. Kuo, “The modified analytic trivialization of singularities”, J. Math. Soc. Japan 32 (1980), p. 605-614 Article | MR 589100 | Zbl 0509.58007
[23] T.-C. Kuo, “On classification of real singularities”, Invent. Math. 82 (1985), p. 257-262 Article | MR 809714 | Zbl 0587.32018
[24] K. Kurdyka, “Ensembles semi-algébriques symétriques par arcs”, Math. Ann. 282 (1988), p. 445-462 Article | MR 967023 | Zbl 0686.14027
[25] K. Kurdyka, “Injective endomorphisms of real algebraic sets are surjective”, Math. Ann. 282 (1998), p. 1-14 MR 1666793 | Zbl 0933.14036
[26] LÊ Dung Tráng, Topologie des singularités des hypersurfaces complexes, Astérisque, 1973, p. 171-182 Zbl 0331.32009
[27] S. Lojasiewicz, Ensembles semi-analytiques, I.H.E.S., 1965
[28] E. Looijenga, Motivic Measures, Séminaire Bourbaki exposé 874, mars 2000 Numdam | Zbl 0996.14011
[29] C. McCrory & A. Parusiński, “Complex monodromy and the topology of real algebraic sets”, Comp. Math. 106 (1997), p. 211-233 Article | MR 1457340 | Zbl 0949.14037
[30] J. Milnor & P. Orlik, “Isolated singularities defined by weighted homogeneous polynomials”, Topology 9 (1970), p. 385-393 Article | MR 293680 | Zbl 0204.56503
[31] T. Nishimura, “Topological invariance of weights for weighted homogeneous singularities”, Kodai Math. J. 9 (1986), p. 188-190 Article | MR 842866 | Zbl 0612.32001
[32] R. Quarez, “Espace des germes d'arcs réels et série de Poincaré d'un ensemble semi-algébrique”, Ann. Inst. Fourier 51 (2001) no. 1, p. 43-67 Cedram | MR 1821067 | Zbl 0967.14037
[33] O. Saeki, “Topological invariance of weights for weighted homogeneous isolated singularities in $\bb C^3$”, Proc. Amer. Math. Soc. 103 (1988), p. 995-999 MR 947679 | Zbl 0656.32009
[34] B. Teissier, Cycles évanescents, sections planes, et conditions de Whitney, Astérisque, 1973, p. 285-362 Zbl 0295.14003
[35] W. Veys, “The topological zeta function associated to a function on a normal surface germ”, Topology 38 (1999), p. 439-456 Article | MR 1660317 | Zbl 0947.32020
[36] S.-T. Yau, “Topological types and multiplicity of isolated quasihomogeneous surface singularities”, Bull. Amer. Math. Soc. 19 (1988), p. 447-454 Article | MR 935021 | Zbl 0659.32013
[37] E. Yoshinaga & M. Suzuki, “Topological types of quasihomogeneous singularities in $\bb C^2$”, Topology 18 (1979), p. 113-116 Article | MR 544152 | Zbl 0428.32004
Satoshi Koike; Adam Parusiński
Motivic-type invariants of blow-analytic equivalence
(Invariants de type motivique de l'équivalence blow-analytique)
Annales de l'institut Fourier, 53 no. 7 (2003), p. 2061-2104, doi: 10.5802/aif.2001
Article PDF | Reviews MR 2044168 | Zbl 1062.14006 | 1 citation in Cedram
Class. Math.: 14B05, 32S15
Keywords: blow-analytic equivalence, motivic integration, zeta functions, Thom-Sebastiani formulae
Résumé - Abstract
Bibliography
[2] E. Bierstone & P.D. Milman, “Arc-analytic functions”, Invent. Math. 101 (1990), p. 411-424 Article | MR 1062969 | Zbl 0723.32005
[3] E. Bierstone & P.D. Milman, “Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant”, Invent. Math. 128 (1997), p. 207-302 Article | MR 1440306 | Zbl 0896.14006
[4] J. Damon & T. Gaffney, “Topological triviality of deformations of functions and Newton filtrations”, Invent. Math. 72 (1983), p. 335-358 Article | MR 704395 | Zbl 0519.58021
[5] J. Denef & F. Loeser, “Motivic Igusa zeta functions”, J. Alg. Geom. 7 (1998), p. 505-537 MR 1618144 | Zbl 0943.14010
[6] J. Denef & F. Loeser, “Germs of arcs on singular algebraic varieties and motivic integration”, Invent. Math. 135 (1999), p. 201-232 Article | MR 1664700 | Zbl 0928.14004
[7] J. Denef & F. Loeser, “Motivic exponential integrals and a motivic Thom-Sebastiani Theorem”, Duke Math. J. 99 (1999), p. 289-309 Article | MR 1708026 | Zbl 0966.14015
[8] J. Denef & F. Loeser, Geometry of arc spaces of algebraic varieties, 2001, p. 327-348 Zbl 01944722
[9] J. Denef & F. Loeser, “Lefschetz numbers of iterates of the monodromy and truncated arcs”, Topology 41 (2002), p. 1031-1040 Article | MR 1923998 | Zbl 1054.14003
[10] T. Fukui & E. Yoshinaga, “The modified analytic trivialization of family of real analytic functions”, Invent. Math. 82 (1985), p. 467-477 Article | MR 811547 | Zbl 0559.58005
[11] T. Fukui, “Seeking invariants for blow-analytic equivalence”, Comp. Math. 105 (1997), p. 95-107 Article | MR 1436747 | Zbl 0873.32008
[12] T. Fukui, S. Koike & T.-C. Kuo, Blow-analytic equisingularities, properties, problems and progress, Real Analytic and Algebraic Singularities, Pitman Research Notes in Math. Series, 1998, p. 8-29 Zbl 0954.26012
[13] T. Fukui & L. Paunescu, “Modified analytic trivialization for weighted homogeneous function-germs”, J. Math. Soc. Japan 52 (2000), p. 433-446 Article | MR 1742795 | Zbl 0964.32023
[14] J.-P. Henry & A. Parusiński, “Existence of Moduli for bi-Lipschitz equivalence of analytic functions”, Comp. Math. 136 (2003), p. 217-235 Article | MR 1967391 | Zbl 1026.32055
[15] J.-P. Henry & A. Parusiński, “Invariants of bi-Lipschitz equivalence of real analytic functions”, Banach Center Publications (to appear) MR 2104338 | Zbl 1059.32006
[16] H. Hironaka, “Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II”, Ann. of Math. 79 (1964), p. 109-302 Article | MR 199184 | Zbl 0122.38603
[17] S. Izumi, S. Koike & T.-C. Kuo, “Computations and Stability of the Fukui Invariant”, Comp. Math. 130 (2002), p. 49-73 Article | MR 1883691 | Zbl 1007.58023
[18] M. Kontsevich, “”, Lecture at Orsay, December 7, 1995
[19] W. Kucharz, “Examples in the theory of sufficiency of jets”, Proc. Amer. Math. Soc. 96 (1986), p. 163-166 Article | MR 813830 | Zbl 0594.58008
[20] N. Kuiper, $C^1$-equivalence of functions near isolated critical points, Annales of Math. Studies, Princeton Univ. Press, 1972, p. 199-218 Zbl 0236.58001
[21] T.-C. Kuo, “On $C^0$-sufficiency of jets of potential functions”, Topology 8 (1969), p. 167-171 Article | MR 238338 | Zbl 0183.04601
[22] T.-C. Kuo, “The modified analytic trivialization of singularities”, J. Math. Soc. Japan 32 (1980), p. 605-614 Article | MR 589100 | Zbl 0509.58007
[23] T.-C. Kuo, “On classification of real singularities”, Invent. Math. 82 (1985), p. 257-262 Article | MR 809714 | Zbl 0587.32018
[24] K. Kurdyka, “Ensembles semi-algébriques symétriques par arcs”, Math. Ann. 282 (1988), p. 445-462 Article | MR 967023 | Zbl 0686.14027
[25] K. Kurdyka, “Injective endomorphisms of real algebraic sets are surjective”, Math. Ann. 282 (1998), p. 1-14 MR 1666793 | Zbl 0933.14036
[26] LÊ Dung Tráng, Topologie des singularités des hypersurfaces complexes, Astérisque, 1973, p. 171-182 Zbl 0331.32009
[27] S. Lojasiewicz, Ensembles semi-analytiques, I.H.E.S., 1965
[28] E. Looijenga, Motivic Measures, Séminaire Bourbaki exposé 874, mars 2000 Numdam | Zbl 0996.14011
[29] C. McCrory & A. Parusiński, “Complex monodromy and the topology of real algebraic sets”, Comp. Math. 106 (1997), p. 211-233 Article | MR 1457340 | Zbl 0949.14037
[30] J. Milnor & P. Orlik, “Isolated singularities defined by weighted homogeneous polynomials”, Topology 9 (1970), p. 385-393 Article | MR 293680 | Zbl 0204.56503
[31] T. Nishimura, “Topological invariance of weights for weighted homogeneous singularities”, Kodai Math. J. 9 (1986), p. 188-190 Article | MR 842866 | Zbl 0612.32001
[32] R. Quarez, “Espace des germes d'arcs réels et série de Poincaré d'un ensemble semi-algébrique”, Ann. Inst. Fourier 51 (2001) no. 1, p. 43-67 Cedram | MR 1821067 | Zbl 0967.14037
[33] O. Saeki, “Topological invariance of weights for weighted homogeneous isolated singularities in $\bb C^3$”, Proc. Amer. Math. Soc. 103 (1988), p. 995-999 MR 947679 | Zbl 0656.32009
[34] B. Teissier, Cycles évanescents, sections planes, et conditions de Whitney, Astérisque, 1973, p. 285-362 Zbl 0295.14003
[35] W. Veys, “The topological zeta function associated to a function on a normal surface germ”, Topology 38 (1999), p. 439-456 Article | MR 1660317 | Zbl 0947.32020
[36] S.-T. Yau, “Topological types and multiplicity of isolated quasihomogeneous surface singularities”, Bull. Amer. Math. Soc. 19 (1988), p. 447-454 Article | MR 935021 | Zbl 0659.32013
[37] E. Yoshinaga & M. Suzuki, “Topological types of quasihomogeneous singularities in $\bb C^2$”, Topology 18 (1979), p. 113-116 Article | MR 544152 | Zbl 0428.32004
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