Semi-algebraic neighborhoods of closed semi-algebraic sets

Dutertre, Nicolas

doi:10.5802/aif.2435

Semi-algebraic neighborhoods of closed semi-algebraic sets
[Voisinages semi-algébriques d’ensembles semi-algébriques fermés]

Dutertre, Nicolas

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 429-458.

Résumé
Abstract

Étant donné un ensemble semi-algébrique fermé (non nécessairement compact) $X$ de $ℝ^{n}$ , nous construisons une fonction semi-algébrique $f$ positive et de classe $𝒞^{2}$ telle que $X = f^{- 1} (0)$ et telle que pour $δ > 0$ suffisamment petit, l’inclusion de $X$ dans $f^{- 1} ([0, δ])$ soit une rétraction. En corollaire, nous obtenons plusieurs formules pour la caractéristique d’Euler de $X$ .

Given a closed (not necessarly compact) semi-algebraic set $X$ in $ℝ^{n}$ , we construct a non-negative semi-algebraic $𝒞^{2}$ function $f$ such that $X = f^{- 1} (0)$ and such that for $δ > 0$ sufficiently small, the inclusion of $X$ in $f^{- 1} ([0, δ])$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$ .

MR 2514870 | Zbl 1174.14051

DOI : https://doi.org/10.5802/aif.2435
Classification : 14P10, 14P25
Mots clés : voisinage tubulaire, ensembles semi-algébriques, rétraction, function semi-algébrique approchante quasirégulière, voisinage semi-algébrique approchant quasirégulier

@article{AIF_2009__59_1_429_0,
     author = {Dutertre, Nicolas},
     title = {Semi-algebraic neighborhoods of closed~semi-algebraic sets},
     journal = {Annales de l'Institut Fourier},
     pages = {429--458},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2435},
     mrnumber = {2514870},
     zbl = {1174.14051},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2435/}
}

Dutertre, Nicolas. Semi-algebraic neighborhoods of closed semi-algebraic sets. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 429-458. doi : 10.5802/aif.2435. http://archive.numdam.org/articles/10.5802/aif.2435/

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