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

Two proper polynomial maps $f_1,f_2\text{\hspace{0.17em}⁣}:{\text{mC}}^n\text{lr}{\text{mC}}^n$ are said to be \emph{equivalent} if there exist ${\mathrm{\Phi }}_1,{\mathrm{\Phi }}_2\in \text{Aut}({\text{mC}}^n)$ such that $f_2={\mathrm{\Phi }}_2\circ f_1\circ {\mathrm{\Phi }}_1$. In this article we investigate proper polynomial maps of topological degree $d\ge 2$ 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.