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Zbl 1174.14051
Dutertre, Nicolas
Semi-algebraic neighborhoods of closed semi-algebraic sets. (English)
[J] Ann. Inst. Fourier 59, No. 1, 429-458 (2009). ISSN 0373-0956; ISSN 1777-5310/e

The author extends some results of {\it A. H. Durfee} on the approximation of compact semi-algebraic sets [see Trans. Am. Math. Soc. 276, 517--530 (1983; Zbl 0529.14013)], to the noncompact case. \par The main result of this paper is the following: Given a closed, possibly noncompact, semi-algebraic set $X$ in ${\mathbb R}^n$, it is possible to construct a non-negative semi-algebraic $C^2$ function $f$ such that $X = f^{-1}(0)$ and such that for $\delta > 0$ sufficiently small, the inclusion of $X$ in $f^{-1}[-\delta,\delta]$ is a retraction. \par As an application of this result the author gives some degree formulas for the Euler-Poincaré characteristic of any closed semi-algebraic set and a Petrovskii-Oleinik inequality for the Euler-Poincaré characteristic of any real algebraic set.
[Antonio Diaz-Cano (Madrid)]

MSC 2000:: *14P10 Semialgebraic sets and related spaces
14P25 Topology of real algebraic varieties

Keywords: tubular neighborhood; semi-algebraic sets; regular approaching semi-algebraic neighborhood

Citations: Zbl 0529.14013

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