On \(p\)-parameter bifurcation of an \(n\)-dimensional function-germ
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.
scientific article; zbMATH DE number 1138671