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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 \(k\).

58K05 Critical points of functions and mappings on manifolds
14P10 Semialgebraic sets and related spaces
57R35 Differentiable mappings in differential topology
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