Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Examples in the theory of sufficiency of jets


Author: Wojciech Kucharz
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] V. I. Arnol'd, Normal forms in neighborhoods of degenerate critical points, Russian Math. Surveys 29 (1974), 10-50.
  • [2] J. Bochnak and W. Kucharz, Sur les germs d'applications différentiables à singularités isolées, Trans. Amer. Math. Soc. 252 (1979), 115-131. MR 534113 (80j:58009)
  • [3] J. Bochnak and T. C. Kuo, Different realizations of nonsufficient jet, Indag. Math. 34 (1972), 24-31. MR 0307267 (46:6387)
  • [4] J. Bochnak and S. Lojasiewicz, A converse of the Kuiper-Kuo theorem, Lecture Notes in Math., vol. 192, Springer-Verlag, Berlin and New York, 1971, pp. 254-261. MR 0291971 (45:1059)
  • [5] S. H. Chang and Y. C. Lu, On $ {C^0}$-sufficiency of complex jets, Canad. J. Math. 25 (1973), 874-880. MR 0343310 (49:8052)
  • [6] T. Fukuda, Types topologiques des polynômes, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 87-106. MR 0494152 (58:13080)
  • [7] H. King, Topological type in families of germs, Invent. Math. 62 (1980), 1-13. MR 595579 (83a:58014)
  • [8] S. Koike and W. Kucharz, Sur les réalisations de jets non-suffisants, C. R. Acad. Sci. Paris Sér. I Math. 288 (1979). MR 527696 (80g:58004)
  • [9] T. C. Kuo, On $ {C^0}$-sufficiency of jets of potential functions, Topology 8 (1969), 167-171. MR 0238338 (38:6614)
  • [10] -, Characterizations of $ V$-sufficiency of jets, Topology 11 (1972), 115-131. MR 0288775 (44:5971)
  • [11] N. Kuiper, $ {C^1}$-equivalence of functions near isolated critical points, Sympos. Infinite Dimensional Topology (Baton Rouge, 1967), Princeton Univ. Press, Princeton, N. J.; Univ. of Tokyo Press, Tokyo, 1972, pp. 199-218. MR 0413161 (54:1282)
  • [12] D. T. Lê and C. P. Ramanujam, The invariance of Milnor's number implies the invariance of topological type, Amer. J. Math. 98 (1976), 67-78. MR 0399088 (53:2939)
  • [13] J. Milnor, Singular points of complex hypersurfaces, Ann of Math. Stud., vol. 61, Princeton Univ. Press, Princeton, N.J., 1968. MR 0239612 (39:969)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 58C27, 57R45

Retrieve articles in all journals with MSC: 58C27, 57R45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0813830-1
Article copyright: © Copyright 1986 American Mathematical Society

American Mathematical Society