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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.
number of branches, 32B30, Differentiable maps on manifolds, topological degrees, Theory of singularities and catastrophe theory, 32B20, real analytic map, 58C27
number of branches, 32B30, Differentiable maps on manifolds, topological degrees, Theory of singularities and catastrophe theory, 32B20, real analytic map, 58C27
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