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

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 $Φ_1, Φ_2 \in \textrm{Aut}(\mathbb{C}^2)$ such that $f_2=Φ_2 \circ f_1 \circ Φ_1$. 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 $d=2$.

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