Linearity and Non-linearity of Groups of Polynomial Automorphisms of $K^2$

TL;DR

This paper explores the linearity properties of subgroups within the polynomial automorphism group of the affine plane over various fields, revealing characteristic-dependent behaviors and subgroup distinctions.

Contribution

It characterizes when subgroups of polynomial automorphisms are linear or nonlinear over different fields, highlighting differences between characteristic zero and finite characteristic cases.

Findings

In characteristic zero, small nonlinear and large linear subgroups exist.

Over finite fields, the entire automorphism group is linear.

For infinite fields of finite characteristic, the group is nonlinear.

Abstract

Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial automorphisms of $K^2$. We investigate which subgroups are linear or not. In characteristic zero, there are small nonlinear subgroups and some big linear subgroups. When $K$ has finite characteristic, the whole group $\Aut\,K^2$ is linear whenever $K$ is finite, and nonlinear otherwise.