Coordinates, retracts and automorphisms

Abstract

Let $K$ be a field of characteristic zero, $K[x,y]$ be the polynomial ring in two variables. Let $\varphi =(f,g)$ be an endomorphism of $K[x,y]$. It is proved that if $\varphi$ maps each coordinate to a generator of some proper retract, then it is an automorphism. As a corollary, the retract preserving problem is solved for both polynomial ring over $K$ and free algebra over an arbitrary field when $n=2$.

7 pages

References in corpus (1)

• Endomorphisms preserving coordinates of polynomial algebras