Mapping degree formula for 2-parameter bifurcation of function-germs
Let \(f\), \(g\) be two analytic functions in \(B_ \varepsilon \subset \mathbb{R}^{n+2}\), where \(B_ \varepsilon\) is a sufficiently small closed ball of radius \(\varepsilon\) and centered at the origin of \(\mathbb{R}^{n+2}\), \((x,y,z_ 1,\dots,z_ n)\in \mathbb{R}^{n+2}\), \(f(0) = g(0) = 0\). Let us denote \(F_ \delta = f^{-1}(\delta) \cap B_ \varepsilon\), \(F_ \delta(g \geq 0) = F_ \delta \cap \{g\geq 0\}\) and \(F_ \delta(g \leq 0) = F_ \delta \cap \{g \leq 0\}\). If we put \(j = \partial(g,f)/\partial(x,y)\), then we can define the map \(H_{g,f} : \mathbb{R}^{n+2} \to \mathbb{R}^{n+2}\) by the relation \(H_{g,f}(x,y,z_ 1,\dots,z_ n) = (f(\cdot),j(\cdot),f_{z_ 1}(\cdot),\dots,f_{z_ n}(\cdot))\). Under certain special conditions on the function \(g\) the author proved the following degree formula for the mapping \(H: \text{deg }H = - \text{sign}(-\delta)^ n\{\chi(F_ \delta(g \geq 0)) - \chi(F_ \delta(g\leq 0))\}\), where \(\text{deg }H\) is the Brouwer degree of \(H\) and \(\chi(X)\) denotes the Euler characteristic of \(X\).
scientific article; zbMATH DE number 488113