Proper polynomial self-maps of the affine space: state of the art and new results

TL;DR

This paper studies proper polynomial maps in complex affine spaces, focusing on their classification up to automorphism equivalence, presenting recent results for two-dimensional cases and extending some findings to higher dimensions.

Contribution

It provides new insights into the classification of proper polynomial maps, especially in two dimensions, and extends some results to higher-dimensional cases.

Findings

Classification results for proper polynomial maps in 2

Partial extension of classification to higher dimensions

Descriptions of recent advances in the field

Abstract

Two proper polynomial maps $f_1, \,f_2 \colon \mC^n \lr \mC^n$ are said to be \emph{equivalent} if there exist $\Phi_1,\, \Phi_2 \in \textrm{Aut}(\mC^n)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. In this article we investigate proper polynomial maps of topological degree $d \geq 2$ up to equivalence. In particular we describe some of our recent results in the case $n=2$ and we partially extend them in higher dimension.