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

Abstract

Two proper polynomial maps $f_1,f_2:{\mathbb{C}}^2⟶{\mathbb{C}}^2$ are said to be \emph{equivalent} if there exist ${\Phi }_1,{\Phi }_2\in \text{Aut}({\mathbb{C}}^2)$ such that $f_2={\Phi }_2\circ f_1\circ {\Phi }_1$. We investigate proper polynomial maps of arbitrary topological degree $d\ge 2$ 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 $d=2$.

arXiv comment15 pages. Final version, to appear in Journal of Geometric Analysis