Examples in the theory of sufficiency of jets
HTML articles powered by AMS MathViewer
- by Wojciech Kucharz PDF
- Proc. Amer. Math. Soc. 96 (1986), 163-166 Request permission
Abstract:
It is shown that for a given nonnegative integer $s$, there exist a positive integer $r(s)$ and an $r(s)$-jet ${\upsilon _s}$ with source at $O \in {{\mathbf {R}}^3}$ 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 $O \in {{\mathbf {C}}^2}$ which is $V$-sufficient but not ${C^0}$-sufficient in the class of holomorphic realizations is constructed.References
-
V. I. Arnol’d, Normal forms in neighborhoods of degenerate critical points, Russian Math. Surveys 29 (1974), 10-50.
- Jacek Bochnak and Wojciech Kucharz, Sur les germes d’applications differentiables à singularités isolées, Trans. Amer. Math. Soc. 252 (1979), 115–131 (French). MR 534113, DOI 10.1090/S0002-9947-1979-0534113-4
- 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. MR 0307267
- J. Bochnak and S. Łojasiewicz, A converse of the Kuiper-Kuo theorem, Proceedings of Liverpool Singularities—Symposium, I (1969/70), Lecture Notes in Mathematics, Vol. 192, Springer, Berlin, 1971, pp. 254–261. MR 0291971
- S. H. Chang and Y. C. Lu, On $C^{0}$-sufficiency of complex jets, Canadian J. Math. 25 (1973), 874–880. MR 343310, DOI 10.4153/CJM-1973-091-0
- Takuo Fukuda, Types topologiques des polynômes, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 87–106 (French). MR 494152
- Henry C. King, Topological type in families of germs, Invent. Math. 62 (1980/81), no. 1, 1–13. MR 595579, DOI 10.1007/BF01391660
- 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). MR 527696
- Tzee Char Kuo, On $C^{0}$-sufficiency of jets of potential functions, Topology 8 (1969), 167–171. MR 238338, DOI 10.1016/0040-9383(69)90007-X
- Tzee Char Kuo, Characterizations of $v$-sufficiency of jets, Topology 11 (1972), 115–131. MR 288775, DOI 10.1016/0040-9383(72)90026-2
- Nicolaas H. Kuiper, $C^{1}$-equivalence of functions near isolated critical points, Symposium on Infinite-Dimensional Topology (Louisiana State Univ., Baton Rouge, La., 1967) Ann. of Math. Studies, No. 69, Princeton Univ. Press, Princeton, N. J., 1972, pp. 199–218. MR 0413161
- 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. MR 399088, DOI 10.2307/2373614
- 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. MR 0239612
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 96 (1986), 163-166
- MSC: Primary 58C27; Secondary 57R45
- DOI: https://doi.org/10.1090/S0002-9939-1986-0813830-1
- MathSciNet review: 813830