Semi-algebraic neighborhoods of closed semi-algebraic sets

Nicolas Dutertre (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités

Given a closed (not necessarily compact) semi-algebraic set $X$ in $\mathbb{R}^n$, we construct a nonnegative semi-algebraic $C^2$ function $f$ such that $X=f^{-1}(0)$ and such that for $\delta > 0$ sufficiently small, the inclusion $X \subset f^{-1}([0,\delta])$ is a retraction. As a corollary, we obtain several formulas for $\chi(X)$.

Domaines: Géométrie algébrique [math.AG]

Nicolas Dutertre. Semi-algebraic neighborhoods of closed semi-algebraic sets. Annales de l'Institut Fourier, 2009, 59 (1), pp.429-458. ⟨hal-00018914⟩