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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
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