An elementay proof of the automorphism theorem for the polynomial ring in two variables (Q1109821)

The authors have given an elementary proof of the following theorem: If \(\phi\) is a \({\mathbb{C}}\)-algebra automorphism of \({\mathbb{C}}[x,y]\) with \(\phi (x)=f(x,y)\), \(\phi (y)=g(x,y)\), \(\deg (f(0,y))=n\geq 1\), \(\deg (g(0,y))=m\geq 1\), then \(m| n\) or \(n| m\). From this they easily deduce that every \({\mathbb{C}}\)-algebra automorphism is a finite (compositional) product of automorphisms of the following types: \((i)\quad x\mapsto \lambda_{11}x+\lambda_{12}y+\lambda_{13},\) \(y\mapsto \lambda_{21}x+\lambda_{22}y+\lambda_{23}\) with \(\lambda_{11}\lambda_{22}-\lambda_{12}\lambda_{21}\neq 0\) and \(\lambda_{ij}\in {\mathbb{C}}\); \((ii)\quad x\mapsto x,\) \(y\mapsto y+h(x)\) with h(x)\(\in {\mathbb{C}}[x]\).