Semi-algebraic neighborhoods of closed semi-algebraic sets

Nicolas Dutertre[1]

  • [1] Université de Provence Centre de Mathématiques et Informatique 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 1, page 429-458
  • ISSN: 0373-0956

Abstract

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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 .

How to cite

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Dutertre, Nicolas. "Semi-algebraic neighborhoods of closed semi-algebraic sets." Annales de l’institut Fourier 59.1 (2009): 429-458. <http://eudml.org/doc/10398>.

@article{Dutertre2009,
abstract = {Given a closed (not necessarly compact) semi-algebraic set $X$ in $\mathbb\{R\}^n$, we construct a non-negative semi-algebraic $\{\mathcal\{C\}\}^2$ function $f$ such that $\{X\{=\}f^\{-1\}(0)\}$ and such that for $\delta &gt;0$ sufficiently small, the inclusion of $X$ in $f^\{-1\}([0,\delta ])$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.},
affiliation = {Université de Provence Centre de Mathématiques et Informatique 39 rue Joliot-Curie 13453 Marseille Cedex 13 (France)},
author = {Dutertre, Nicolas},
journal = {Annales de l’institut Fourier},
keywords = {Tubular neighborhood; semi-algebraic sets; retraction; quasiregular approaching semi-algebraic function; quasiregular approaching semi-algebraic neighborhood; tubular neighborhood; regular approaching semi-algebraic neighborhood},
language = {eng},
number = {1},
pages = {429-458},
publisher = {Association des Annales de l’institut Fourier},
title = {Semi-algebraic neighborhoods of closed semi-algebraic sets},
url = {http://eudml.org/doc/10398},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Dutertre, Nicolas
TI - Semi-algebraic neighborhoods of closed semi-algebraic sets
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 429
EP - 458
AB - Given a closed (not necessarly compact) semi-algebraic set $X$ in $\mathbb{R}^n$, we construct a non-negative semi-algebraic ${\mathcal{C}}^2$ function $f$ such that ${X{=}f^{-1}(0)}$ and such that for $\delta &gt;0$ sufficiently small, the inclusion of $X$ in $f^{-1}([0,\delta ])$ is a retraction. As a corollary, we obtain several formulas for the Euler characteristic of $X$.
LA - eng
KW - Tubular neighborhood; semi-algebraic sets; retraction; quasiregular approaching semi-algebraic function; quasiregular approaching semi-algebraic neighborhood; tubular neighborhood; regular approaching semi-algebraic neighborhood
UR - http://eudml.org/doc/10398
ER -

References

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