A non-tame and non-co-tame automorphism of the polynomial ring

TL;DR

This paper presents the first example of a polynomial ring automorphism in three variables that is neither tame nor co-tame, expanding understanding of automorphism classifications in algebra.

Contribution

It constructs the first known automorphism of this type by using the exponential automorphism of a locally nilpotent derivation of rank three.

Findings

Provides the first example of a non-tame, non-co-tame automorphism in three variables.

Uses exponential automorphism of a locally nilpotent derivation to construct the example.

Enhances the classification of polynomial ring automorphisms.

Abstract

An automorphism $F$ of the polynomial ring in $n$ variables over a field of characteristic zero is said to be {\it co-tame} if the subgroup of the automorphism group of the polynomial ring generated by $F$ and affine automorphisms contains the tame subgroup. There exist many examples of such an $F$, and several sufficient conditions for co-tameness are already known. In 2015, Edo-Lewis gave the first example of a non-cotame automorphism, which is a tame automorphism of the polynomial ring in three variables. In this paper, we give the first example of a non-cotame automorphism which is not tame. We construct such an example when $n=3$ as the exponential automorphism of a locally nilpotent derivation of rank three.