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 -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
Posted:
October 27, 2006
MathSciNet review:
2262910
Full-text PDF Free Access
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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  -bi-Lipschitz equivalence, is finite.
- 1.
S. Alvarez, L. Birbrair, J. Costa, A. Fernandes, Topological
 -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
(91j:14045)
- 3.
Riccardo
Benedetti and Masahiro
Shiota, Finiteness of semialgebraic types of polynomial
functions, Math. Z. 208 (1991), no. 4,
589–596. MR 1136477
(92j:14068), http://dx.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 (58 #13080)
- 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
(2004d:32037), http://dx.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
(43 #1215a)
- 8.
James
A. Montaldi, On contact between submanifolds, Michigan Math.
J. 33 (1986), no. 2, 195–199. MR 837577
(87i:58024), http://dx.doi.org/10.1307/mmj/1029003348
- 9.
Tadeusz
Mostowski, Lipschitz equisingularity, Dissertationes Math.
(Rozprawy Mat.) 243 (1985), 46. MR 808226
(87e:32008)
- 10.
Isao
Nakai, On topological types of polynomial mappings, Topology
23 (1984), no. 1, 45–66. MR 721451
(85g:58076), http://dx.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
(90e:32016)
- 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
(85g:58021)
- 13.
René
Thom, La stabilité topologique des applications
polynomiales, Enseignement Math. (2) 8 (1962),
24–33 (French). MR 0148079
(26 #5588)
- 1.
- S. Alvarez, L. Birbrair, J. Costa, A. Fernandes, Topological -equivalence of analytic function-germs, preprint (2004).
- 2.
- R. Benedetti, J-J. Risler, Real algebraic and semi-algebraic sets. Actualités Mathématiques. Hermann, Paris, 1990. 340 pp. MR 1070358 (91j:14045)
- 3.
- R. Benedetti, M. Shiota, Finiteness of semialgebraic types of polynomial functions. Math Z. 208 (1991), no. 4, 589-596. MR 1136477 (92j:14068)
- 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.
- T. Fukuda, Types topologiques des polynômes, Inst. Hautes Études Sci. Publ. Math. no. 46 (1976), 87-106. MR 0494152 (58:13080)
- 6.
- J.-P. Henry, A. Parusinski, Existence of moduli for bi-Lipschitz equivalence of analytic functions, Compositio Math. 136 (2003), no. 2, 217-235. MR 1967391 (2004d:32037)
- 7.
- J. Mather, Stability of
 -mappings, III: finitely determined map germs, Publ. Math. I.H.E.S. 35 (1969), 127-156. MR 0275459 (43:1215a)
- 8.
- J. Montaldi, On contact between submanifolds, Michigan Math. Journal 33 (1986), 195-199. MR 0837577 (87i:58024)
- 9.
- T. Mostowski, Lipschitz equisingularity, Dissertationes Math. 243 (1985). MR 0808226 (87e:32008)
- 10.
- I. Nakai, On topological types of polynomial mappings. Topology 23 (1984), no. 1, 45-66. MR 0721451 (85g:58076)
- 11.
- A. Parusinski, Lipschitz properties of semi-analytic sets, Ann. Inst. Fourier (Grenoble) 38 (1988), no. 4, 189-213. MR 0978246 (90e:32016)
- 12.
- C. Sabbah, Le type topologique éclaté d'une application analytique. Singularities, Part 2 (Arcata, Calif., 1981), 433-440, Proc. Sympos. Pure Math., 40, Amer. Math. Soc., Providence, RI, 1983. MR 0713269 (85g:58021)
- 13.
- R. Thom, La stabilité topologique des applications polynomiales. Enseignement Math. (2) 8 (1962), 24-33. MR 0148079 (26:5588)
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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:
http://dx.doi.org/10.1090/S0002-9939-06-08566-2
PII:
S 0002-9939(06)08566-2
Received by editor(s):
May 14, 2005
Received by editor(s) in revised form:
November 4, 2005
Posted:
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
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