On stable polynomial mappings

TL;DR

This paper characterizes topologically stable polynomial mappings from ^2 to ^2 with bounded degrees, providing methods to identify generic mappings and understand their stability under small deformations.

Contribution

It offers a characterization of topologically stable polynomial mappings in ^2 with degree bounds and methods to determine generic mappings within this class.

Findings

Characterization of topologically stable polynomial mappings.

Method to effectively identify generic mappings.

Insights into stability under small deformations.

Abstract

For given natural numbers $d_1,d_2$ let $\Omega_2(d_1,d_2)$ be the set off all polynomial mappings $F=(f,g):\mathbb{C}^2\to\mathbb{C}^2$ such that deg $f\le d_1$, deg $g\le d_2$. We say that the mapping $F$ is topologically stable in $\Omega_2(d_1,d_2)$ if for every small deformation $F_t\in \Omega_2(d_1,d_2)$ the mapping $F_t$ is topologically equivalent to the mapping $F$. The aim of this paper is to characterize the topologically stable mappings in $\Omega_2(d_1,d_2)$. In particular we show how to effectively determine a member of $\Omega_2(d_1,d_2)$ with generic topology.