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

Abstract

Denote by $H(d_1,d_2,d_3)$ the set of all homogeneous polynomial mappings $F=(f_1,f_2,f_3): \C^3\to\C^3$, such that $°f_i=d_i$. We show that if $\gcd (d_i,d_j)\le 2$ for $1\le i<j\le 3$ and $\gcd (d_1,d_2,d_3)=1$, then there is a non-empty Zariski open subset $U\subset H(d_1,d_2,d_3)$ such that for every mapping $F\in U$ the map germ $(F,0)$ is $\mathcal{A}$-finitely determined. Moreover, in this case we compute the number of discrete singularities ($0$-stable singularities) of a generic mapping $(f_1,f_2,f_3):\C^3\to\C^3$, where $°f_i=d_i$.