We study the topological K-equivalence of function-germs (ℝn, 0) → (ℝ, 0). We present some special classes of piece-wise linear functions and prove that they are normal forms for equivalence classes with respect to topological K-equivalence for definable functions-germs. For the case n = 2 we present polynomial models for analytic function-germs.
@article{bwmeta1.element.doi-10_2478_s11533-010-0013-8,
author = {S\'ergio Alvarez and Lev Birbrair and Jo\~ao Costa and Alexandre Fernandes},
title = {Topological K-equivalence of analytic function-germs},
journal = {Open Mathematics},
volume = {8},
year = {2010},
pages = {338-345},
zbl = {1226.32014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0013-8}
}
Sérgio Alvarez; Lev Birbrair; João Costa; Alexandre Fernandes. Topological K-equivalence of analytic function-germs. Open Mathematics, Tome 8 (2010) pp. 338-345. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-010-0013-8/
[1] Birbair L., Costa J., Fernandes A., Ruas M., K-bi-Lipschitz equivalence of real function-germs, Proc. Amer. Math. Soc, 2007, 135(4), 1089–1095 http://dx.doi.org/10.1090/S0002-9939-06-08566-2 | Zbl 1116.32020
[2] Birbair L, Costa J., Fernandes A., Finiteness theorem for topological contact equivalence of map germs, Hokkaido Math. J., 2009, 38(3), 511–517
[3] Bierstone E., Milman P., Uniformization of analytic spaces, J. Amer. Math. Soc, 1989, 2(4), 801–836 http://dx.doi.org/10.2307/1990895 | Zbl 0685.32007
[4] Benedetti R., Shiota M., Finiteness of semialgebraic types of polynomial functions, Math. Z., 1991, 208(4), 589–596 http://dx.doi.org/10.1007/BF02571547 | Zbl 0744.14034
[5] Coste M., An introduction to 0-minimal geometry, PhD thesis, University of Pisa, Italy, 2000 (in Italian)
[6] Fukuda T., Types topolodiques des polynomes, Inst. Hautes Etudes Sci. Publ. Math., 1976(46), 87–106 | Zbl 0341.57019
[7] Nishimura T., Topological K-equivalence of smooth map-germs, Stratifications, singularities and differential equations, I, (Marseille, 1990; Honolulu, HI, 1990), 82–93, Travaux en Cours, 54, Hermann, Paris, 1997
[8] Nishimura T, C 0-K-determined map-germs, Trans. Amer. Math. Soc, 1989, 132(2), 621–639 54, Hermann, Paris 1997. http://dx.doi.org/10.2307/2001003
[9] Prishlyak A., Topological equivalence of smooth functions with isolated critical points on a closed surface, Topology and Applications, 2002, 119(3), 257–267 http://dx.doi.org/10.1016/S0166-8641(01)00077-3 | Zbl 1042.57021
[10] van den Dries L, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, 248, Cambridge University Press, Cambridge, 1998
[11] Ruas M., Valette G., C o and bi-Lipschitz K-equivalence of mappings, preprint
[12] Wall C.T.C., Finite determinacy of smooth map-germs, Bull. London Math. Soc, 1981, 13, 481–539 http://dx.doi.org/10.1112/blms/13.6.481 | Zbl 0451.58009