On the topological invariants of germs of analytic functions
Let f: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) be a germ of a real analytic function. Let \(f_ C\) be its complexification and let e(f) denote the Euler characteristic of the set \(f_ C^{-1}(\alpha)\cap B_ r^{2n}\) where \(0<\alpha \ll r\ll 1\), \(B_ r^{2n}=\{z\in C^ n|\| z\| <r\}\). If \(f_ C\) has an isolated critical point at 0, it follows from results of a paper of \textit{C. T. C. Wall} [ibid. 22, 345-350 (1983; Zbl 0516.58010)] that e(f) mod 2 is a topological invariant of f. In the paper under review the author proves that e(f) mod 2 is a topological invariant for any germ of a real analytic function. This result follows from a more general result which is proven in the paper: For each integer \(d\geq 2\) the author defines a certain property \(A_ d\) for real analytic function germs. (For example homogeneous polynomials of degree d have property \(A_ d.)\) Furthermore, all analytic germs f: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) are defined to have property \(A_ 1\). The main result of the paper is that for \(d\geq 1\), e(f)/d mod 2 is a topological invariant for analytic germs f: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) having property \(A_ d.\) For each \(d\geq 2\) the author also gives examples of homogeneous polynomials f and g of degree d such that e(f)/d\(\not\equiv e(g)/d mod 2\). So f and g are consequently not topologically right equivalent. However, if d is even then e(f)\(\equiv e(g)\equiv 0 mod 2\). e(f)/d mod 2 is therefore a more precise topological invariant than e(f) mod 2.
scientific article; zbMATH DE number 4028453