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On \(p\)-parameter bifurcation of an \(n\)-dimensional function-germ. (English) Zbl 0902.32015

Let \(f,g : (\mathbb{R}^{n+p},0) \to (\mathbb{R},0)\) be analytic germs with \(0\) an isolated singularity of \(f\) and grad \(g(0) \neq 0\). For \(\varepsilon > 0\) and \(| \delta| \) small set \(F_\delta := f^{-1}(\delta) \cap B_\varepsilon(0)\). The author computes the difference of Euler characteristics, \(\chi(F_\delta \cap \{g\geq 0\})-\chi(F_\delta \cap \{g \leq 0\})\), in terms of the degree of an associated map \(H=H(g,f) : B_\varepsilon (0) \to \mathbb{R}^{n+p}\), provided \(H^{-1}(0)=0\). The components of \(H\) are given by \(f\) and certain products of the first partial derivatives of \(f\) and \(g\). The proof uses arguments from Morse theory.

MSC:

32S05 Local complex singularities
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:

[1] Fukui, T., Mapping degree formula for 2-parameter bifurcation of function-germs, Topology, 32, 567-571 (1993) · Zbl 0789.58023 · doi:10.1016/0040-9383(93)90007-I
[2] MILNOR, J.: Morse theory,Ann. Math. Stud.51, Princeton University Press (1963). · Zbl 0108.10401
[3] Szafraniec, Z., A formula for the Euler characteristic of a real algebraic manifold, manuscripta mathematica, 85, 345-360 (1993) · Zbl 0824.14046 · doi:10.1007/BF02568203
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