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

Fukui, Toshizumi; Koike, Satoshi; Shiota, Masahiro

Annales de l'Institut Fourier, Tome 48 (1998) no. 5, pp. 1395-1440.

Dans ce travail nous introduisons la notion utile de trivialité modifiée de Nash d’une famille d’ensembles de zéros de germes d’applications polynomiales réelles. Nous donnons d’abord un lemme d’isotopie de Nash permettant d’obtenir la trivialité. Ensuite, à l’aide de ceci, nous montrons deux types de théorèmes de trivialité modifiée de Nash et un théorème de classification finie pour la trivialité.

Ces théorèmes renforcent des résultats topologiques similaires.

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.

[1] M. Artin and B. Mazur, On periodic points, Ann. of Math., 81 (1965), 82-99. | MR | Zbl

[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. | MR | Zbl

[3] M. Buchner and W. Kucharz, Topological triviality of a family of zero-sets, Proc. Amer. Math. Soc., 102 (1988), 699-705. | MR | Zbl

[4] M. Coste and M. Shiota, Nash triviality in families of Nash manifolds, Invent. Math., 108 (1992), 349-368. | MR | Zbl

[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. | MR | Zbl

[6] J. Damon, Topological invariants of µ-constant deformations of complete intersection singularities, Quart. J. Math., 40 (1989), 139-159. | MR | Zbl

[7] J. Damon, Topological triviality and versality for subgroup of A and K : II. Sufficient conditions and applications, Nonlinearity, 5 (1992), 373-412. | MR | Zbl

[8] V. I. Danilov, The geometry of toric varieties, Russ. Math. Surveys, 33 (1978), 97-154. | MR | Zbl

[9] V.I. Danilov, Newton polyhedra and vanishing cohomology, Funct. Anal. Appl., 13 (1979), 103-115. | MR | Zbl

[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. | Zbl

[11] T. Fukuda, Types topologiques des polynômes, Publ. Math. IHES, 46 (1976), 87-106. | Numdam | MR | Zbl

[12] T. Fukuda, Topological triviality of real analytic singularities, preprint.

[13] T. Fukui, The modified analytic trivialization of a family of real analytic mappings, Contemporary Math., 90 (1989), 73-89. | MR | Zbl

[14] T. Fukui, Modified analytic trivialization via weighted blowing up, J. Math. Soc. Japan, 44 (1992), 455-459. | MR | Zbl

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

[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. | MR | Zbl

[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. | Zbl

[18] A. G. Khovanskii, Newton polyhedra and toroidal varieties, Funct. Anal. Appl., 11 (1977), 289-295. | MR | Zbl

[19] A.G. Khovanskii, Newton polyhedra and the genus of complete intersections, Funct. Anal. Appl., 12 (1978), 38-46. | MR | Zbl

[20] H. King, Topological type in families of germs, Invent. Math., 62 (1980), 1-13. | MR | Zbl

[21] S. Koike, On strong C0-equivalence of real analytic functions, J. Math. Soc. Japan, 45 (1993), 313-320. | Zbl

[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. | MR | Zbl

[23] A.G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math., 32 (1976), 1-31. | MR | Zbl

[24] T.C. Kuo, Une classification des singularités réelles, C. R. Acad. Sci. Paris, 288 (1979), 809-812. | MR | Zbl

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

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

[27] S. Lojasiewicz, Triangulation of semi-analytic sets, Ann. Scu. Norm. di Pisa, 18 (1964), 449-474. | Numdam | MR | Zbl

[28] B. Malgrange, Ideals of differentiable functions, Oxford Univ. Press, 1966.

[29] M. Oka, On the bifurcation of the multiplicity and topology of the Newton boundary, J. Math. Soc. Japan, 31 (1979), 435-450. | MR | Zbl

[30] M. Oka, On the weak simultaneous resolution of a negligible truncation of the Newton boundary, Contemporary Mathematics, 90 (1989), 199-210. | MR | Zbl

[31] M. Oka, Non-degenerate complete intersection singularities, Actualités Mathématiques, Hermann, 1997. | Zbl

[32] A. Seidenberg, A new decision method for elementary algebra, Ann. of Math., 60 (1954), 365-374. | MR | Zbl

[33] M. Shiota, Classification of Nash manifolds, Ann. Inst. Fourier, 33-3 (1983), 209-232. | Numdam | MR | Zbl

[34] M. Shiota, Nash manifolds, Lect. Notes in Math. 1269, Springer, 1987. | MR | Zbl

[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. | Zbl

[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. | MR | Zbl

[37] T. Fukui and L. Paunescu, Modified analytic trivialization for weighted homogeneous function-germs, preprint. | Zbl