Journal of the Mathematical Society of Japan

Blow-Nash types of simple singularities

Goulwen FICHOU

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We address the question of the classification under blow-Nash equivalence of simple Nash function germs. We state that this classification coincides with the real analytic classification. We prove moreover that a simple germ can not be blow-Nash equivalent to a nonsimple one. The method is based on the computation of relevant coefficients of the real zeta functions associated to a Nash germ via motivic integration.

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J. Math. Soc. Japan Volume 60, Number 2 (2008), 445-470.

First available in Project Euclid: 30 May 2008

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Zentralblatt MATH identifier

Primary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]
Secondary: 14P20: Nash functions and manifolds [See also 32C07, 58A07] 14P25: Topology of real algebraic varieties 32S15: Equisingularity (topological and analytic) [See also 14E15]

blow-Nash equivalence simple singularities virtual Poincaré polynomial


FICHOU, Goulwen. Blow-Nash types of simple singularities. J. Math. Soc. Japan 60 (2008), no. 2, 445--470. doi:10.2969/jmsj/06020445.

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  • V. Arnold, S. Goussein-Zadé and A. Varchenko, Singularity of differentiable maps, Birkhauser, Boston, 1985.
  • J. Denef and F. Loeser, Germs of arcs on singular algebraic varieties and motivic integration, Invent. Math., 135 (1999), 201–232.
  • G. Fichou, Motivic invariants of arc-symmetric sets and blow-Nash equivalence, Compositio Math., 141 (2005), 655–688.
  • G. Fichou, The corank and index are blow-Nash invariants, Kodai Math. J., 29 (2006), 31–40.
  • T. Fukui, T. C. Kuo and L. Paunescu, Constructing blow-analytic isomorphisms, Ann. Inst. Fourier (Grenoble), 51 (2001), 1071–1087.
  • T. Fukui and L. Paunescu, Modified analytic trivialization for weighted homogeneous function-germs, J. Math. Soc. Japan, 52 (2000), 433–446.
  • T. Fukui and L. Paunescu, On blow-analytic equivalence, to appear In: Arc Spaces and Additive Invariants in Real Algebraic Geometry, Proceedings of Winter School, Real algebraic and Analytic Geometry and Motivic Integration, Aussois, 2003, (eds. M. Coste, K. Kurdyka and A. Parusinski), Panoramas et Synthèses, SMF.
  • S. Izumi, S. Koike and T. C. Kuo, Computations and stability of the Fukui invariant, Compositio Math., 130 (2002), 49–73.
  • S. Koike, Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous polynomial mappings, J. Math. Soc. Japan, 49 (1997), 617–631.
  • T. C. Kuo, On classification of real singularities, Invent. Math., 82 (1985), 257–262.
  • C. McCrory and A. Parusiński, Virtual Betti numbers of real algebraic varieties, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 763–768.
  • L. Paunescu, Invariants associated with blow-analytic homeomorphisms, Proc. Japan Acad., Ser. A, Math. Sci., 78 (2002), 194–198.
  • M. Shiota, Nash manifolds, Lect. Notes in Math., 1269, Springer-Verlag, 1987.