On the number of branches of an 1-dimensional semianalytic set
Let \(F=(F_ 1,...,F_{n-1}): ({\mathbb{R}}^ n,0)\to ({\mathbb{R}}^{n-1},0)\) be a germ of an analytic map and assume that 0 is an isolated singular point of \(F^{-1}(0)=X\). Let b(F) be the number of branches of X-\(\{\) \(0\}\). Let G: (\({\mathbb{R}}^ n,0)\to ({\mathbb{R}},0)\) be another analytic germ and suppose that 0 is isolated in \(G^{-1}(0)\cap X\). Let \(b_+(G,F)\) (resp. \(b_-(G,F))\) be the number of branches of X-\(\{\) \(0\}\) where G is positive (resp. negative). Letting \(\Delta\) be the Jacobean of (G,F), \(H=(\Delta,F)\) and taking suitable representatives of the above map-germs the author proves that \(b_+(G,F)-b_-(G,F)=2 \deg (H).\) Choosing \(G=x^ 2_ 1+...+x^ 2_ n\), a formula for b(F) is obtained. If \(X\cap \{x_ n=0\}=\{0\},\) a formula for the difference between the number of branches of X-\(\{\) \(0\}\) contained in \(\{x_ n>0\}\) and the number of branches contained in \(\{x_ n<0\}\) is also obtained. These two formulas have also been proved in recent papers of Aoki, Fukuda, Sun and Nishimura. Finally, the author considers a more general class of 1- dimensional semianalytic sets, and it is shown how the number of branches of such sets can be computed in terms of the topological degrees of some finite families of map-germs.
scientific article; zbMATH DE number 4064090