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)

 
 

 

Infinite determinacy on a closed set for smooth germs with non-isolated singularities


Author: Vincent Thilliez
Journal: Proc. Amer. Math. Soc. 134 (2006), 1527-1536
MSC (2000): Primary 58K40
DOI: https://doi.org/10.1090/S0002-9939-05-08253-5
Published electronically: October 13, 2005
MathSciNet review: 2199202
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give necessary and sufficient conditions for infinite determinacy of a smooth function germ whose critical locus contains a given set. This set is assumed to be the zero variety $ X $ of some analytic map-germ having maximal rank on a dense subset of $ X $. We obtain a result in terms of \Lojasiewicz estimates which extends, in particular, previous works by Sun and Wilson on line singularities, and by Grandjean on singularities of codimension $ 1 $ or $ 2 $.


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

  • 1. J. Bochnak and J.-J. Risler, Sur les exposants de \Lojasiewicz, Comment. Math. Helvetici 50 (1975), 493-507. MR 0404674 (53:8474)
  • 2. D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Graduate Texts in Math. 150, Springer-Verlag (1995). MR 1322960 (97a:13001)
  • 3. V. Grandjean, Infinite relative determinacy of smooth function germs with transverse isolated singularities and relative \Lojasiewicz conditions, J. London Math. Soc. 69 (2004), 518-530. MR 2040619 (2005a:58060)
  • 4. S. Izumiya and S. Matsuoka, Notes on smooth function germs on varieties, Proc. Amer. Math. Soc. 97 (1986), 146-150. MR 0831404 (87g:58013)
  • 5. L. Kushner and B. Terra Leme, Finite relative determination and relative stability, Pacific J. Math. 192 (2000), 315-328. MR 1744573 (2000m:58069)
  • 6. B. Malgrange, Ideals of Differentiable Functions, Oxford University Press (1965).
  • 7. T.C. Nguyen, H.D. Nguyen, S.M. Nguyen, and H.V. Ha, Sur les germes de fonctions infiniment déterminés, C. R. Acad. Sci. Paris 285 (1977), 1045-1048. MR 0649266 (58:31180)
  • 8. D. Siersma, Isolated line singularities, Proc. Symp. Pure Math. 40 (1983), 485-495. MR 0713274 (85d:32017)
  • 9. B. Sun and L.C. Wilson, Determinacy of smooth germs with real isolated line singularities, Proc. Amer. Math. Soc. 129 (2001), 2789-2797. MR 1838804 (2002d:58050)
  • 10. J.-C. Tougeron, Idéaux de fonctions différentiables, Springer-Verlag (1972). MR 0440598 (55:13472)
  • 11. C.T.C. Wall, Finite determinacy of smooth map-germs, Bull. London Math. Soc. 13 (1981), 481-539. MR 0634595 (83i:58020)
  • 12. L.C. Wilson, Infinitely determined mapgerms, Can. J. Math. 33 (1981), 671-684. MR 0627650 (82k:58023)
  • 13. L.C. Wilson, personal communication (October 2003).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58K40

Retrieve articles in all journals with MSC (2000): 58K40


Additional Information

Vincent Thilliez
Affiliation: Mathématiques - Bâtiment M2, Université des Sciences et Technologies de Lille, F-59655 Villeneuve d’Ascq Cedex, France
Email: thilliez@math.univ-lille1.fr

DOI: https://doi.org/10.1090/S0002-9939-05-08253-5
Received by editor(s): December 13, 2004
Published electronically: October 13, 2005
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society