Remarks on retracts of polynomial rings in three variables in any characteristic

Abstract

Let $A$ be a retract of the polynomial ring in three variables over a field $k$. It is known that if ${\rm char}\: (k) = 0$ or ${\rm tr.deg}\:_k A \not= 2$ then $A$ is a polynomial ring. In this paper, we give some sufficient conditions for $A$ to be the polynomial ring in two variables over $k$ when ${\rm char}\: (k) > 0$ and ${\rm tr.deg}\:_k A = 2$.