The Preimage of a Coordinate

Abstract

Let K be a field of characteristic zero. Based on the degree estimate of Makar-Limanov and Yu, we prove that the preimage of a coordinate under an injective endomorphism of K〈x, y〉 is also a coordinate. As by-products, we give new proofs of the following results: (1) the preimage of a coordinate under an injective endomorphism of K[x,y] is also a coordinate; (2) any automorphism of K[x,y] or K〈x, y〉 is tame.