𝒦-bi-Lipschitz equivalence of real function-germs

Birbrair, L.; Costa, J.; Fernandes, A.; Ruas, M.

doi:10.1090/S0002-9939-06-08566-2

$\mathcal{K}$ -bi-Lipschitz equivalence of real function-germs

Authors: L. Birbrair, J. C. F. Costa, A. Fernandes and M. A. S. Ruas
Journal: Proc. Amer. Math. Soc. 135 (2007), 1089-1095
MSC (2000): Primary 32S15, 32S05
DOI: https://doi.org/10.1090/S0002-9939-06-08566-2
Published electronically: October 27, 2006
MathSciNet review: 2262910
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove that the set of equivalence classes of germs of real polynomials of degree less than or equal to , with respect to $\mathcal{K}$ -bi-Lipschitz equivalence, is finite.

1. S. Alvarez, L. Birbrair, J. Costa, A. Fernandes, Topological $\mathcal{K}$ -equivalence of analytic function-germs, preprint (2004).
2. Riccardo Benedetti and Jean-Jacques Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358
3. Riccardo Benedetti and Masahiro Shiota, Finiteness of semialgebraic types of polynomial functions, Math. Z. 208 (1991), no. 4, 589–596. MR 1136477, https://doi.org/10.1007/BF02571547
4. M. Coste, An introduction to o-minimal geometry. Dottorato di Ricerca in Matematica, Dip. Mat. Univ. Pisa. Instituti Editoriali e Poligrafici Internazionali 2000.
5. Takuo Fukuda, Types topologiques des polynômes, Inst. Hautes Études Sci. Publ. Math. 46 (1976), 87–106 (French). MR 0494152
6. Jean-Pierre Henry and Adam Parusiński, Existence of moduli for bi-Lipschitz equivalence of analytic functions, Compositio Math. 136 (2003), no. 2, 217–235. MR 1967391, https://doi.org/10.1023/A:1022726806349
7. John N. Mather, Stability of 𝐶^{∞} mappings. III. Finitely determined mapgerms, Inst. Hautes Études Sci. Publ. Math. 35 (1968), 279–308. MR 0275459
John N. Mather, Stability of 𝐶^{∞} mappings. IV. Classification of stable germs by 𝑅-algebras, Inst. Hautes Études Sci. Publ. Math. 37 (1969), 223–248. MR 0275460
John N. Mather, Stability of 𝐶^{∞} mappings. V. Transversality, Advances in Math. 4 (1970), 301–336 (1970). MR 0275461, https://doi.org/10.1016/0001-8708(70)90028-9
8. James A. Montaldi, On contact between submanifolds, Michigan Math. J. 33 (1986), no. 2, 195–199. MR 837577, https://doi.org/10.1307/mmj/1029003348
9. Tadeusz Mostowski, Lipschitz equisingularity, Dissertationes Math. (Rozprawy Mat.) 243 (1985), 46. MR 808226
10. Isao Nakai, On topological types of polynomial mappings, Topology 23 (1984), no. 1, 45–66. MR 721451, https://doi.org/10.1016/0040-9383(84)90024-7
11. Adam Parusiński, Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 4, 189–213 (English, with French summary). MR 978246
12. C. Sabbah, Le type topologique éclaté d’une application analytique, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 433–440 (French). MR 713269
13. René Thom, La stabilité topologique des applications polynomiales, Enseignement Math. (2) 8 (1962), 24–33 (French). MR 0148079

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 32S15, 32S05

Retrieve articles in all journals with MSC (2000): 32S15, 32S05

Additional Information

L. Birbrair
Affiliation: Departamento de Matemàtica, Universidade Federal do Cearà, Av. Mister Hull s/u Campus do PICI, Bloco 914, CEP 60, 455-760 Fortaleza-CE, Brazil

J. C. F. Costa
Affiliation: Departamento de Matemàtica (IBILCE), Universidade Estadual Paulista, Sao Jose de Rio Preto, SP 15054-000 Brazil

A. Fernandes
Affiliation: Departamento de Matemàtica, Universidade Federal do Cearà, Av. Mister Hull s/u Campus do PICI, Bloco 914, CEP 60, 455-760 Fortaleza-CE, Brazil

M. A. S. Ruas
Affiliation: Institute of Sciences and Mathematics, University of Sao Paulo, Sao Carlos SP, Brazil

DOI: https://doi.org/10.1090/S0002-9939-06-08566-2
Received by editor(s): May 14, 2005
Received by editor(s) in revised form: November 4, 2005
Published electronically: October 27, 2006
Additional Notes: The first named author was supported by CNPq grant No. 300985/93-2.
The second named author was supported by Fapesp grant No. 01/14577-0.
The fourth named author was supported by CNPq grant No. 301474/2005-2.
Communicated by: Mikhail Shubin
Article copyright: © Copyright 2006 American Mathematical Society