Generic $\mathcal{A}$-finite determinacy and singularities of homogeneous polynomial mappings

TL;DR

This paper investigates the generic properties of polynomial mappings, extending previous results on homogeneous polynomial mappings from dimension 3 to higher dimensions, with some limitations.

Contribution

It generalizes key findings on $ ext{A}$-finite determinacy and singularities of homogeneous polynomial mappings to dimensions ≥2, expanding the scope of prior work.

Findings

Extended generic properties to higher dimensions

Proved $ ext{A}$-finite determinacy in broader contexts

Identified dimensional limitations for property extensions

Abstract

We make a detailed investigation of the generic properties that polynomial mappings possess. An important starting point is the work by Farnik, Jelonek and Ruas in 2019, where they prove some of those properties in the context of homogeneous polynomial mappings of $\mathbb{C}^3$ to $\mathbb{C}^3$, and conclude the genericity of $\mathcal{A}$-finite determinacy by applying the geometric criterion. Using their strategy, we further extend and generalize some of their key findings to dimensions greater than or equal to $2$, though some of those properties can only be extended up to dimension $4$.