Finite \(\mathcal{A}\)-determinacy of generic homogeneous map germs in \(\mathbb{C}^3\) (Q1996212)

Let \(\Omega (d_{1},d_{2},d_{3})\) (resp. \(H(d_{1},d_{2},d_{3}))\) be the set of polynomial (resp. homogeneous) mappings \(F=(f_{1},f_{2},f_{3}):\mathbb{C} ^{3}\rightarrow \mathbb{C}^{3}\) such that \(\deg f_{i}=d_{i}.\) The main result is that if \(\operatorname{GCD}(d_{i},d_{j})\leq 2\) for \(1\leq i<j\leq 3\) and \(\operatorname{GCD}(d_{1},d_{2},d_{3})=1\) then there exists a non-empty Zariski open set \(U\subset H(d_{1},d_{2},d_{3})\) such that for every maping \( F\in U\) the local map-germ \((F,0)\) at \(0\) is finitely \(\mathcal{A}\)-determined. In the remaining cases there are no finitely \(\mathcal{A}\)-determined map-germs. As an application the authors compute (in the first case) the number of discrete singularities of a generic mapping in \(\Omega (d_{1},d_{2},d_{3}).\)