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

TL;DR

This paper proves that generic homogeneous polynomial mappings in three complex variables are finitely determined under certain degree conditions and computes the number of their discrete singularities.

Contribution

It establishes conditions under which generic homogeneous map germs in \\mathbb{C}^3 are finitely determined and calculates the number of discrete singularities for these mappings.

Findings

Existence of a non-empty Zariski open subset where map germs are \\mathcal{A}-finitely determined.

Explicit computation of the number of discrete singularities for generic mappings.

Conditions on degrees d_i ensuring finite determinacy and singularity count.

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 $\deg f_i=d_i$. We show that if $\gcd(d_i,d_j)\leq 2$ for $1\leq i<j\leq 3$ 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 $\deg f_i=d_i$.