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On Efroymson’s extenson theorem for Nash functions. (English) Zbl 0581.14016

Let \(\Omega\) be a semi-algebraic open subset of \({\mathbb{R}}^ n\), and let v be a Nash function on \(\Omega\) and \(V=v^{-1}(0)\neq \emptyset\). The author proves that if \(f_ 1\) is a Nash function defined in a semi- algebraic neighborhood of V, then here exists a Nash function g on \(\Omega\), such that \(f_ 1-g=kv\), where k is a Nash function defined in a semi-algebraic neighborhood of V. This main result of the paper is a little more precise form of a theorem of G. A. Efroymson [cf. Géométrie algébrique réelle et formes quadratiques, Journ. S. M. F., Univ. Rennes 1981, Lect. Notes Math. 959, 343-357 (1982; Zbl 0516.14020)]. The author considers his proof as elementary. Probably this depends on the readers background: the author uses the theory of the real spectrum instead of triangulations.
Reviewer: N.V.Ivanov

MSC:

14Pxx Real algebraic and real-analytic geometry
58A07 Real-analytic and Nash manifolds
32C05 Real-analytic manifolds, real-analytic spaces

Citations:

Zbl 0516.14020
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References:

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