The automorphism theorem and additive group actions on the affine plane

TL;DR

This paper provides a new, concise proof of the theorem that the invariant ring for additive group actions on the affine plane is generated by a single coordinate, utilizing the automorphism theorem of Jung and van der Kulk.

Contribution

It introduces a simplified proof of a known result about invariant rings under additive group actions using classical automorphism theorems.

Findings

Invariant ring for ${f G}_a$-actions is generated by one coordinate

New proof is shorter and relies on automorphism theorems

Connects automorphism groups with invariant theory

Abstract

Due to Rentschler, Miyanishi and Kojima, the invariant ring for a ${\bf G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk.