Determinacy of smooth germs with real isolated line singularities
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- by Bohao Sun and Leslie C. Wilson PDF
- Proc. Amer. Math. Soc. 129 (2001), 2789-2797 Request permission
Abstract:
The germ of a smooth real-valued function on Euclidean space is called a real isolated line singularity if its singular set is a nonsingular curve, its Jacobian ideal is Łojasiewicz at the singular set, and its Hessian determinant restricted to the singular set is Łojasiewicz at 0. Consider the set of all germs whose singular set contains a fixed nonsingular curve $L$. We prove that such a germ $f$ is infinitely determined among all such germs with respect to composition by diffeomorphisms preserving $L$ if, and only if, the Jacobian ideal of $f$ contains all germs which vanish on $L$ and are infinitely flat at 0 if, and only if, $f$ is a real isolated line singularity.References
- Shyūichi Izumiya and Sachiko Matsuoka, Notes on smooth function germs on varieties, Proc. Amer. Math. Soc. 97 (1986), no. 1, 146–150. MR 831404, DOI 10.1090/S0002-9939-1986-0831404-3
- Jean Martinet, Singularities of smooth functions and maps, London Mathematical Society Lecture Note Series, vol. 58, Cambridge University Press, Cambridge-New York, 1982. Translated from the French by Carl P. Simon. MR 671585
- John N. Mather, Stability of $C^{\infty }$ mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 279–308. MR 275459
- Ruud Pellikaan, Finite determinacy of functions with nonisolated singularities, Proc. London Math. Soc. (3) 57 (1988), no. 2, 357–382. MR 950595, DOI 10.1112/plms/s3-57.2.357
- Dirk Siersma, Isolated line singularities, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 485–496. MR 713274, DOI 10.1090/pspum/040.2/713274
- Jean-Claude Tougeron, Idéaux de fonctions différentiables, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 71, Springer-Verlag, Berlin-New York, 1972. MR 0440598
- C. T. C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), no. 6, 481–539. MR 634595, DOI 10.1112/blms/13.6.481
- Leslie C. Wilson, Infinitely determined map germs, Canadian J. Math. 33 (1981), no. 3, 671–684. MR 627650, DOI 10.4153/CJM-1981-053-3
Additional Information
- Bohao Sun
- Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
- Leslie C. Wilson
- Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
- Email: les@math.hawaii.edu
- Received by editor(s): January 5, 2000
- Published electronically: February 9, 2001
- Communicated by: Jozef Dodziuk
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2789-2797
- MSC (1991): Primary 58K40; Secondary 32S05
- DOI: https://doi.org/10.1090/S0002-9939-01-06068-3
- MathSciNet review: 1838804