Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous polynomial mappings

Satoshi KOIKE

doi:10.2969/jmsj/04930617

Satoshi KOIKE

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JOURNAL FREE ACCESS

1997 Volume 49 Issue 3 Pages 617-631

DOI https://doi.org/10.2969/jmsj/04930617

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Published: 1997 Received: July 13, 1994 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: SUBTITLE Details: Wrong : Dedicated to Professor Yasutoshi Nomura on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) J. Briancon and J. P. Speder, La trivialité topologique n'implique pas les conditions de Whitney, C. R. Acad. Sci. Paris, 280 (1975), 365-367.
2) M. Buchner and W. Kucharz, Topologic al triviality of a family of zero-sets, Proc. Amer. Math. Soc., 102 (1988), 699-705.
3) M. Coste and M. Shiota, Thom's first isotopy lemma: a semialgebraic version with uniform bounds, Real Analytic and Algebraic Geometry (ed. F. Broglia, M. Galbiati and A. Tognoli), Walter de Gruyter, 1995, 83-101.
4) T. Fukui, Toric resolution and real function, preprint.
5) H. King, Topological type in families of germs, Invent. Math., 62 (1980), 1-13.
6) S. Koike, On strong C⁰-equivalence of real analytic functions, J. Math. Soc. Japan, 45 (1993), 313-320.
7) S. Koike, Modified Nash triviality of a family of zero-sets of weighted homogeneous polynomial mappings, Proceedings of International Workshop on Topology and Geometry at Hanoi, Kodai J. Math., 17 (1994), 432-437.
8) S. Koike, Strong C⁰-triviality of family of weighted homogeneous polynomials (in preparation).
9) T. C. Kuo, On C⁰-sufficiency of jets of potential functions, Topology, 8 (1969), 167-171.
10) T. C. Kuo, Characterizations of v-sufficiency of jets, Topology, 11 (1972), 115-131.
11) T. C. Kuo, Une classification des singularités réelles, C.R. Acad. Sci. Paris, 288 (1979), 809-812.
12) T. C. Kuo, The modified analytic trivialization of singularities, J. Math. Soc. Japan, 32 (1980), 605-614.
13) T.C. Kuo, On classification of real singularities, Invent. Math., 82 (1985), 257-262.
14) B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, 1966.
15) M. Shiota, Classification of Nash manifolds, Ann. Inst. Fourier, 33 (1983), 209-232.
16) M. Shiota, Nash manifolds, Lect. Notes in Math., 1269, Springer, 1987.
17) H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math., 66 (1957), 545-556.

Right : [1] J. Briançon and J. P. Speder, La trivialité topologique n'implique pas les conditions de Whitney, C. R. Acad. Sci. Paris, 280 (1975), 365-367.
[2] M. Buchner and W. Kucharz, Topologic al triviality of a family of zero-sets, Proc. Amer. Math. Soc., 102 (1988), 699-705.
[3] M. Coste and M. Shiota, Thom's first isotopy lemma: a semialgebraic version with uniform bounds, Real Analytic and Algebraic Geometry (ed. F. Broglia, M. Galbiati and A. Tognoli), Walter de Gruyter, 1995, 83-101.
[4] T. Fukui, Toric resolution and real function, preprint.
[5] H. King, Topological type in families of germs, Invent. Math., 62 (1980), 1-13.
[6] S. Koike, On strong C⁰-equivalence of real analytic functions, J. Math. Soc. Japan, 45 (1993), 313-320.
[7] S. Koike, Modified Nash triviality of a family of zero-sets of weighted homogeneous polynomial mappings, Proceedings of International Workshop on Topology and Geometry at Hanoi, Kodai J. Math., 17 (1994), 432-437.
[8] S. Koike, Strong C⁰-triviality of family of weighted homogeneous polynomials (in preparation).
[9] T. C. Kuo, On C⁰-sufficiency of jets of potential functions, Topology, 8 (1969), 167-171.
[10] T. C. Kuo, Characterizations of v-sufficiency of jets, Topology, 11 (1972), 115-131.
[11] T. C. Kuo, Une classification des singularités réelles, C.R. Acad. Sci. Paris, 288 (1979), 809-812.
[12] T. C. Kuo, The modified analytic trivialization of singularities, J. Math. Soc. Japan, 32 (1980), 605-614.
[13] T. C. Kuo, On classification of real singularities, Invent. Math., 82 (1985), 257-262.
[14] B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, 1966.
[15] M. Shiota, Classification of Nash manifolds, Ann. Inst. Fourier, 33 (1983), 209-232.
[16] M. Shiota, Nash manifolds, Lect. Notes in Math., 1269, Springer, 1987.
[17] H. Whitney, Elementary structure of real algebraic varieties, Ann. of Math., 66 (1957), 545-556.

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