Modified Nash triviality of a family of zero-sets of real polynomial mappings

Toshizumi Fukui; Satoshi Koike; Masahiro Shiota

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 5, page 1395-1440
  • ISSN: 0373-0956

Abstract

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In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.

How to cite

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Fukui, Toshizumi, Koike, Satoshi, and Shiota, Masahiro. "Modified Nash triviality of a family of zero-sets of real polynomial mappings." Annales de l'institut Fourier 48.5 (1998): 1395-1440. <http://eudml.org/doc/75324>.

@article{Fukui1998,
abstract = {In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.},
author = {Fukui, Toshizumi, Koike, Satoshi, Shiota, Masahiro},
journal = {Annales de l'institut Fourier},
keywords = {modified Nash triviality; resolution; toric modification; zero sets of real polynomial map-germs; Nash isotopy lemma},
language = {eng},
number = {5},
pages = {1395-1440},
publisher = {Association des Annales de l'Institut Fourier},
title = {Modified Nash triviality of a family of zero-sets of real polynomial mappings},
url = {http://eudml.org/doc/75324},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Fukui, Toshizumi
AU - Koike, Satoshi
AU - Shiota, Masahiro
TI - Modified Nash triviality of a family of zero-sets of real polynomial mappings
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 5
SP - 1395
EP - 1440
AB - In this paper we introduce the notion of modified Nash triviality for a family of zero sets of real polynomial map-germs as a desirable one. We first give a Nash isotopy lemma which is a useful tool to show triviality.Then, using it, we prove two types of modified Nash triviality theorem and a finite classification theorem for this triviality. These theorems strengthen similar topological results.
LA - eng
KW - modified Nash triviality; resolution; toric modification; zero sets of real polynomial map-germs; Nash isotopy lemma
UR - http://eudml.org/doc/75324
ER -

References

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  1. [1] M. ARTIN and B. MAZUR, On periodic points, Ann. of Math., 81 (1965), 82-99. Zbl0127.13401MR31 #754
  2. [2] 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. Zbl0331.32010MR54 #13122
  3. [3] M. BUCHNER and W. KUCHARZ, Topological triviality of a family of zero-sets, Proc. Amer. Math. Soc., 102 (1988), 699-705. Zbl0658.58011MR89b:58029
  4. [4] M. COSTE and M. SHIOTA, Nash triviality in families of Nash manifolds, Invent. Math., 108 (1992), 349-368. Zbl0801.14017MR93e:14066
  5. [5] M. COSTE and M. SHIOTA, Thom's first isotopy lemma : a semialgebraic version with uniform bounds, Real analytic and algebraic geometry (eds. F.Broglia, M.Galbiati and A. Tognoli), Walter de Gruyter (1995), 83-101. Zbl0844.14025MR96i:14047
  6. [6] J. DAMON, Topological invariants of µ-constant deformations of complete intersection singularities, Quart. J. Math., 40 (1989), 139-159. Zbl0724.32019MR90j:32012
  7. [7] J. DAMON, Topological triviality and versality for subgroup of A and K : II. Sufficient conditions and applications, Nonlinearity, 5 (1992), 373-412. Zbl0747.58014MR93f:58020
  8. [8] V. I. DANILOV, The geometry of toric varieties, Russ. Math. Surveys, 33 (1978), 97-154. Zbl0425.14013MR80g:14001
  9. [9] V.I. DANILOV, Newton polyhedra and vanishing cohomology, Funct. Anal. Appl., 13 (1979), 103-115. Zbl0427.14006MR80h:14001
  10. [10] V.I. DANILOV and A.G. KHOVANSKII, Newton polyhedra and an algorithm for computing Hodge-Deligne numbers, Math. USSR-Izv., 29 (1987), 279-298. Zbl0669.14012
  11. [11] T. FUKUDA, Types topologiques des polynômes, Publ. Math. IHES, 46 (1976), 87-106. Zbl0341.57019MR58 #13080
  12. [12] T. FUKUDA, Topological triviality of real analytic singularities, preprint. Zbl1005.32020
  13. [13] T. FUKUI, The modified analytic trivialization of a family of real analytic mappings, Contemporary Math., 90 (1989), 73-89. Zbl0683.58007MR90f:58014
  14. [14] T. FUKUI, Modified analytic trivialization via weighted blowing up, J. Math. Soc. Japan, 44 (1992), 455-459. Zbl0766.58008MR93e:32043
  15. [15] T. FUKUI and E. YOSHINAGA, The modified analytic trivialization of family of real analytic functions, Invent. Math., 82 (1985), 467-477. Zbl0559.58005MR87a:58028
  16. [16] H. HIRONAKA, Resolution of singularities of an algebraic variety over a field of characteristic zero I, II, Ann. of Math., 79 (1964), 109-326. Zbl0122.38603MR33 #7333
  17. [17] H. HIRONAKA, Stratification and flatness, Real and complex singularities (ed. Holm), Nordic summer school/NAVF Symposium in Mathematics Oslo, Auguest 5-25, 1976, Sijthoff & Noordhoff (1977), 199-265. Zbl0424.32004
  18. [18] A. G. KHOVANSKII, Newton polyhedra and toroidal varieties, Funct. Anal. Appl., 11 (1977), 289-295. Zbl0445.14019MR57 #16291
  19. [19] A.G. KHOVANSKII, Newton polyhedra and the genus of complete intersections, Funct. Anal. Appl., 12 (1978), 38-46. Zbl0406.14035MR80b:14022
  20. [20] H. KING, Topological type in families of germs, Invent. Math., 62 (1980), 1-13. Zbl0477.58010MR83a:58014
  21. [21] S. KOIKE, On strong C0-equivalence of real analytic functions, J. Math. Soc. Japan, 45 (1993), 313-320. Zbl0788.32024
  22. [22] 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. Zbl0916.58005MR98i:58033
  23. [23] A.G. KOUCHNIRENKO, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1-31. Zbl0328.32007MR54 #7454
  24. [24] T.C. KUO, Une classification des singularités réelles, C. R. Acad. Sci. Paris, 288 (1979), 809-812. Zbl0404.58013MR80i:32034
  25. [25] T.C. KUO, The modified analytic trivialization of singularities, J. Math. Soc. Japan, 32 (1980), 605-614. Zbl0509.58007MR82d:58012
  26. [26] T.C. KUO, On classification of real singularities, Invent. Math., 82 (1985), 257-262. Zbl0587.32018MR87d:58025
  27. [27] S. LOJASIEWICZ, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449-474. Zbl0128.17101MR30 #3478
  28. [28] B. MALGRANGE, Ideals of differentiable functions, Oxford Univ. Press, 1966. 
  29. [29] M. OKA, On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math. Soc. Japan, 31 (1979), 435-450. Zbl0408.35012MR80h:32018
  30. [30] M. OKA, On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Contemporary Mathematics, 90 (1989), 199-210. Zbl0682.32011MR90h:32030
  31. [31] M. OKA, Non-degenerate complete intersection singularities, Actualités Mathématiques, Hermann, 1997. Zbl0930.14034
  32. [32] A. SEIDENBERG, A new decision method for elementary algebra, Ann. of Math., 60 (1954), 365-374. Zbl0056.01804MR16,209a
  33. [33] M. SHIOTA, Classification of Nash manifolds, Ann. Inst. Fourier, 33-3 (1983), 209-232. Zbl0495.58001MR85b:58004
  34. [34] M. SHIOTA, Nash manifolds, Lect. Notes in Math. 1269, Springer, 1987. Zbl0629.58002MR89b:58011
  35. [35] A.N. VARčENKO, Theorems on the topological equisingularity of families of algebraic varieties and families of polynomial mappings, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 957-1019. Zbl0251.14006
  36. [36] H. WHITNEY, Local properties of analytic varieties, Differential and Combinatorial Topology (ed. S.S.Cairns), A Symposium in Honor of M. Morse, Princeton University Press (1965), 205-244. Zbl0129.39402MR32 #5924
  37. [37] T. FUKUI and L. PAUNESCU, Modified analytic trivialization for weighted homogeneous function-germs, preprint. Zbl0964.32023

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