A new co-tame automorphism of the polynomial ring

TL;DR

This paper introduces a novel co-tame automorphism of the polynomial ring in n variables that is not in the closure of previously known automorphisms satisfying Edo-Lewis's condition, expanding understanding of automorphism subgroups.

Contribution

The paper presents the first example of a co-tame automorphism outside the closure of EL(n), challenging existing classifications and broadening the scope of co-tame automorphisms.

Findings

Identifies a new co-tame automorphism not in EL(n) closure

Shows the existence of co-tame automorphisms beyond known classes

Expands the understanding of automorphism subgroup structures

Abstract

In this paper, we discuss subgroups of the automorphism group of the polynomial ring in n variables over a field of characteristic zero. An automorphism F is said to be co-tame if the subgroup generated by F and affine automorphisms contains the tame subgroup. In 2017, Edo-Lewis gave a sufficient condition for co-tameness of automorphisms. Let EL(n) be the set of all automorphisms satisfying Edo-Lewis's condition. Then, for a certain topology on the automorphism group of the polynomial ring, any element of the closure of EL(n) is co-tame. Moreover, all the co-tame automorphisms previously known belong to the closure of EL(n). In this paper, we give the first example of co-tame automorphisms in n variables which do not belong to the closure of EL(n).