A Simple Proof of Jung's Theorem on Polynomial Automorphisms of $\C^2$

Abstract

The Automorphism Theorem, discovered first by Jung in 1942, asserts that if $k$ is a field, then every polynomial automorphism of $k^2$ is a finite product of linear automorphisms and automorphisms of the form $(x,y)\mapsto(x+p(y), y) $ for $p\in k[y]$. We present here a simple proof for the case $k=\C$ by using Newton-Puiseux expansions.