Volume de tubes autour de singularités. (On the volume of tubes around singularities) (Q1107669)

Let X be a closed analytic set in \({\mathbb{C}}^ n\) of pure dimension d. The author proves that for any compact polydisc P the volume I(X,P,\(\epsilon)\) of the set of points in P having a distance to \(X\cap P\) less than \(\epsilon\) has an asymptotic expansion: \[ I(X,P,\epsilon)\quad \sim \quad \sum_{i\leq 2N}\sum_{a\in F+N}C_{a,i}\epsilon^ a(\log \epsilon)^ i \] when \(\epsilon\) \(\to 0\), with \(F\subset {\mathbb{Q}}^+\) finite. The principal part of this expansion is \((\pi^ d/d!)vol(X\cap P)\cdot \epsilon^{2d}\) and the author computes the next term for plane curves.