A Characterization of Semisimple Plane Polynomial Automorphisms

TL;DR

This paper extends the classical characterization of semisimple elements from linear groups to the group of complex plane polynomial automorphisms, showing that semisimplicity corresponds to Zariski closed conjugacy classes.

Contribution

It proves that an element of the complex plane polynomial automorphism group is semisimple if and only if its conjugacy class is Zariski closed, generalizing a known linear algebra result.

Findings

Semisimple automorphisms have Zariski closed conjugacy classes.

The characterization parallels the linear case for GL_n(C).

Provides a new understanding of the structure of polynomial automorphisms.

Abstract

It is well-known that an element of the linear group ${\rm GL}_n(\C)$ is semisimple if and only if its conjugacy class is Zariski closed. The aim of this paper is to show that the same result holds for the group of complex plane polynomial automorphisms.