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

TL;DR

This paper investigates conditions under which a retract of a three-variable polynomial ring over a field is itself a polynomial ring, focusing on positive characteristic cases with two-dimensional transcendence degree.

Contribution

It provides new sufficient conditions for a retract to be a polynomial ring in two variables over a field of positive characteristic.

Findings

Identifies conditions ensuring retracts are polynomial rings in positive characteristic

Extends known results from characteristic zero to positive characteristic

Provides criteria for the structure of retracts in three-variable polynomial rings

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 ${\rm tr.deg}\:_k A = 2$.