On the number of branches of a plane curve germ
Let f: (\({\mathbb{R}}^ 2,0)\to ({\mathbb{R}},0)\) be a \(C^{\infty}\) function with an isolated critical point 0. The connected components of the intersection \(f^{-1}(0)\) with a small circuit \(S^ 1_{\epsilon}\) at 0 are called branches of \(f^{-1}(0)\). Let \(J_ f\) be \(J_ f=y\frac{\partial f}{\partial x}-x\frac{\partial f}{\partial y}.\) Main theorem. Let \(J_ f\) be a function with isolated critical point at 0. Then the number of branches of \(f^{-1}(0)=2| \deg (f,J_ f)|\), where \(| \deg (f,J_ f)|\) denotes the absolute value of the topological degree of the mapping \((f,J_ f)/\| f,J_ f)\|: S^ 1_{\epsilon}\to S^ 1\). The proof of the main theorem is very simple and based on the following two lemmas. Lemma 2.1. Let f,g: (\({\mathbb{R}}^ 2,0)\to ({\mathbb{R}},0)\) be \(C^{\infty}\) function germs with isolated critical points such that \(f^{-1}(0)\cap g^{-1}(0)=\{0\}\). Suppose that for any two adjacent branches of \(f^{- 1}(0)\), \(g^{-1}(0)\) has an odd number of branches between them. Then the number of branches of \(f^{-1}(0)=2| \deg (f,g)|.\) Lemma 2.2. In conditions of the main theorem \(J_ f^{-1}(0)\) has an odd number of branches between any pair of adjacent branches of \(f^{- 1}(0).\) The last paragraph of the article is devoted to a generalization of the notion of the degree of map germ on the germ: \(({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^ p,0)\), with \(n>p\). The authors define degree as the number of connected components of \(f^{-1}(0)\cap S_{\epsilon}^{n-1}\). The main result of this paragraph is a statement that the homological behavior of f is completely determined by the homology groups of \(f^{- 1}(0)\cap S_{\epsilon}^{n-1}\) and the orientations. This result is a motivation for studying the number of branches.
scientific article; zbMATH DE number 3988296