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Kodai Mathematical Journal
Article . 1988 . Peer-reviewed
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On the number of branches of a $1$-dimensional semianalytic set

On the number of branches of an 1-dimensional semianalytic set
Authors: Szafraniec, Zbigniew;

On the number of branches of a $1$-dimensional semianalytic set

Abstract

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.

Country
Poland
Related Organizations
Keywords

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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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
20
Average
Top 10%
Average
Green
bronze