On Proper Polynomial Maps of $\mathbb{C}^2.$

TL;DR

This paper studies proper polynomial maps of complex two-dimensional space, classifying them up to automorphism equivalence for various degrees, especially focusing on Galois coverings, extending previous work for degree 2.

Contribution

It provides a comprehensive classification of proper polynomial maps of ^2 for all degrees, including a full description of Galois coverings, generalizing earlier results for degree 2.

Findings

Complete classification of proper polynomial maps for arbitrary degree.

Explicit description of equivalence classes for Galois coverings.

Extension of previous degree 2 results to higher degrees.

Abstract

Two proper polynomial maps $f_1, f_2 \colon \mathbb{C}^2 \longrightarrow \mathbb{C}^2$ are said to be \emph{equivalent} if there exist $\Phi_1, \Phi_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=\Phi_2 \circ f_1 \circ \Phi_1$. We investigate proper polynomial maps of arbitrary topological degree $d \geq 2$ up to equivalence. Under the further assumption that the maps are Galois coverings we also provide the complete description of equivalence classes. This widely extends previous results obtained by Lamy in the case $d=2$.