×

zbMATH — the first resource for mathematics

Examples in the theory of sufficiency of jets. (English) Zbl 0594.58008
Summary: It is shown that for a given nonnegative integer s, there exist a positive integer r(s) and an r(s)-jet \(v_ s\) with source at \(O\in {\mathbb{R}}^ 3\) which is not V-sufficient in the class of \(C^{r(s)+s}\)- realizations and is \(C^ 0\)-sufficient in the class of \(C^{r(s)+s+1}\)- realizations. In the complex case, a jet with source at \(O\in {\mathbb{C}}^ 2\) which is V-sufficient but not \(C^ 0\)-sufficient in the class of holomorphic realizations is constructed.

MSC:
58A20 Jets in global analysis
57R45 Singularities of differentiable mappings in differential topology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] V. I. Arnol’d, Normal forms in neighborhoods of degenerate critical points, Russian Math. Surveys 29 (1974), 10-50. · Zbl 0304.57018
[2] Jacek Bochnak and Wojciech Kucharz, Sur les germes d’applications differentiables à singularités isolées, Trans. Amer. Math. Soc. 252 (1979), 115 – 131 (French). · Zbl 0458.58006
[3] Jacek Bochnak and Tzee-char Kuo, Different realizations of a non sufficient jet, Nederl. Akad. Wetensch. Proc. Ser. A 75=Indag. Math. 34 (1972), 24 – 31. · Zbl 0225.58001
[4] J. Bochnak and S. Łojasiewicz, A converse of the Kuiper-Kuo theorem, Proceedings of Liverpool Singularities — Symposium, I (1969/70), Springer, Berlin, 1971, pp. 254 – 261. Lecture Notes in Math., Vol. 192.
[5] S. H. Chang and Y. C. Lu, On \?\(^{0}\)-sufficiency of complex jets, Canad. J. Math. 25 (1973), 874 – 880. · Zbl 0258.58004
[6] Takuo Fukuda, Types topologiques des polynômes, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 87 – 106 (French). · Zbl 0341.57019
[7] Henry C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), no. 1, 1 – 13. · Zbl 0477.58010
[8] Satoshi Koike and Wojciech Kucharz, Sur les réalisations de jets non suffisants, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 8, A457 – A459 (French, with English summary). · Zbl 0404.58002
[9] Tzee Char Kuo, On \?\(^{0}\)-sufficiency of jets of potential functions, Topology 8 (1969), 167 – 171. · Zbl 0183.04601
[10] Tzee Char Kuo, Characterizations of \?-sufficiency of jets, Topology 11 (1972), 115 – 131. · Zbl 0234.58005
[11] Nicolaas H. Kuiper, \?\textonesuperior -equivalence of functions near isolated critical points, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Princeton Univ. Press, Princeton, N. J., 1972, pp. 199 – 218. Ann. of Math. Studies, No. 69.
[12] Lê Dũng Tráng and C. P. Ramanujam, The invariance of Milnor’s number implies the invariance of the topological type, Amer. J. Math. 98 (1976), no. 1, 67 – 78. · Zbl 0351.32009
[13] John Milnor, Singular points of complex hypersurfaces, Annals of Mathematics Studies, No. 61, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1968. · Zbl 0184.48405
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.