We prove that the set of asymptotic critical values of a function definable in an o-minimal structure is finite, even if the structure is not polynomially bounded. As a consequence, the function is a locally trivial fibration over the complement of this set.
@article{bwmeta1.element.bwnjournal-article-apmv75z1p35bwm,
author = {D'Acunto, D.},
title = {Valeurs critiques asymptotiques d'une fonction d\'efinissable dans une structure o-minimale},
journal = {Annales Polonici Mathematici},
volume = {75},
year = {2000},
pages = {35-45},
language = {fr},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv75z1p35bwm}
}
D'Acunto, D. Valeurs critiques asymptotiques d'une fonction définissable dans une structure o-minimale. Annales Polonici Mathematici, Tome 75 (2000) pp. 35-45. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv75z1p35bwm/
[000] [1] J. Bochnak, M. Coste and M. F. Roy, Real Algebraic Geometry, Ergeb. Math. Grenzgeb. 36, Springer, 1998.
[001] [2] L. van den Dries, Tame Topology and o-Minimal Structures, London Math. Soc. Lecture Note Ser. 248, Cambridge Univ. Press, 1988.
[002] [3] L. van den Dries and C. Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), 497-540. | Zbl 0889.03025
[003] [4] K. Kurdyka, On a subanalytic stratification satisfying a Whitney property with exponent 1, in: Real Algebraic Geometry (Rennes, 1991), Lecture Notes in Math. 1524, Springer, 1992, 316-322. | Zbl 0779.32006
[004] [5] K. Kurdyka, On gradients of functions definable in o-minimal structures, Ann. Inst. Fourier (Grenoble) 48 (1998), 769-783. | Zbl 0934.32009
[005] [6] K. Kurdyka, T. Mostowski and A. Parusiński, Gradient conjecture in o-minimal structures, en préparation. | Zbl 1053.37008
[006] [7] K. Kurdyka, P. Orro and S. Simon, Semialgebraic Sard theorem for generalized critical values, preprint, Univ. Savoie, 1999. | Zbl 1067.58031
[007] [8] T. L. Loi and A. Zaharia, Bifurcation sets of functions definable in o-minimal structures, Illinois J. Math. 42 (1998), 449-457. | Zbl 0948.37030
[008] [9] C. Miller, Exponentiation is hard to avoid, Proc. Amer. Math. Soc. 122 (1994), 257-259. | Zbl 0808.03022
[009] [10] R. S. Palais, Lusternik-Schnirelman theory on Banach manifolds, Topology 5 (1966), 115-132. | Zbl 0143.35203
[010] [11] A. Parusiński, On the bifurcation set of complex polynomial with isolated singularities at infinity, Compositio Math. 97 (1995), 369-384. | Zbl 0840.32007
[011] [12] P. J. Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. 146 (1997), 647-691. | Zbl 0919.58003