On the Euler characteristic of analytic and algebraic sets
Let \(f_ 1,...,f_ s: ({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) be germs of real analytic functions. Define \(g(x)= f^ 2_ 1(x) +...+ f_ s^ 2(x) - a| x|^{2k}\) where \(a>0\) and \(k\gg 0\) is an integer. Let \(S_ r\) be the sphere of radius r in \({\mathbb{R}}^ n\). The following is then proved: There exist \(c>0\), \(\alpha >0\), \(r_ 0>0\) such that if \(a<c\), \(k\geq \alpha\), \(0<r<r_ 0\) then (i) g has an isolated singular point at 0; (ii) the Euler characteristic of \(\{x\in S_ r: f_ 1(x)=...=f_ s(x)=0\}\) is \(1-\deg (dg).\) This result is applied to compute the Euler characteristic of an algebraic subset of \({\mathbb{R}}^ n\).
scientific article; zbMATH DE number 3987660