Coordinates, retracts and automorphisms

TL;DR

This paper proves that certain endomorphisms of polynomial rings in two variables over characteristic zero fields are automorphisms if they map coordinates to generators of proper retracts, solving a key retract preserving problem.

Contribution

It establishes a criterion for automorphisms based on the image of coordinates and solves the retract preserving problem in two-variable polynomial rings and free algebras.

Findings

Endomorphisms mapping coordinates to generators of proper retracts are automorphisms.

The retract preserving problem is solved for polynomial rings and free algebras in two variables.

Provides a new characterization of automorphisms in the context of retracts.

Abstract

Let $K$ be a field of characteristic zero, $K[x,y]$ be the polynomial ring in two variables. Let $\phi=(f, g)$ be an endomorphism of $K[x,y]$. It is proved that if $\phi$ 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 $n=2$.