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A generalized Sard theorem on real closed fields. (English) Zbl 1343.58019
The authors prove the following Sard type result : Let $$R$$ be a real closed field. If $$v$$ is a convex subgroup of $$R$$ and $$f : X\rightarrow R^k$$ is a $$C^1$$ semi-algebraic function, with $$X\subset R^n$$ a bounded semi-algebraic manifold, then for any infinitesimal $$z\in v$$ the image $$f(c_z(f))$$ is $$(k,v)$$-thin (Theorem 3.2).
In Theorem 4.3, an interesting application is given : Let $$X$$ be a $$C^1$$ semi-algebraic submanifold of $$\mathbb{R}^n$$ and let $$f : X\rightarrow \mathbb{R}^k$$ be a $$C^1$$ semi-algebraic mapping. Then, the set of asymptotic critical values of $$f$$ has dimension less than $$k$$.

##### MSC:
 58K05 Critical points of functions and mappings on manifolds 14P10 Semialgebraic sets and related spaces 57R35 Differentiable mappings in differential topology
##### Keywords:
semi-algebraic mapping; asymptotic critical value
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##### References:
 [1] Bochnak, Géométrie algébrique réelle (1987) [2] Fischer, O-minimal m-regular stratification, Ann. Pure Appl. Logic 147 (1-2) pp 101– (2007) · Zbl 1125.03029 · doi:10.1016/j.apal.2007.04.002 [3] Hardt, Semi-algebraic local-triviality in semi-algebraic mappings, Amer. J. Math. 102 (2) pp 291– (1980) · Zbl 0465.14012 · doi:10.2307/2374240 [4] T. Kaiser Lebesgue measure theory and integration theory on arbitrary real closed fields · Zbl 1436.03211 [5] Kurdyka, Rennes 1991, Lecture Notes in Mathematics, Vol. 1524 (1992) [6] Kurdyka, Semialgebraic Sard Theorem for generalized critical values, J. Differential Geom. 56 pp 67– (2000) · Zbl 1067.58031 · doi:10.4310/jdg/1090347525 [7] Lion, Théorème de préparation pour les fonctions logarithmico-exponentielles, Ann. Inst. Fourier (Grenoble) 47 (3) pp 859– (1997) · Zbl 0873.32004 · doi:10.5802/aif.1583 [8] J. Maříková M. Shiota Measuring definable sets in o-minimal fields [9] Rabier, Ehresmanns fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. (2) 146 pp 647– (1997) · Zbl 0919.58003 · doi:10.2307/2952457 [10] Valette, Vanishing homology, Sel. Math. New Series 16 pp 267– (2010) · Zbl 1264.14075 · doi:10.1007/s00029-010-0020-4
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