The tame automorphism group in two variables over basic Artinian rings

TL;DR

This paper explores the conditions under which polynomial automorphisms over Artinian rings are tame, extending classical results to more general rings and providing examples of non-tame automorphisms in positive characteristic.

Contribution

It demonstrates that the Q-algebra condition is essential for tameness over Artinian rings and offers a complete description of tame automorphisms for certain local Artinian rings.

Findings

Tameness over Artinian rings requires the Q-algebra assumption.

Complete description of tame automorphisms for local Artinian rings with square-zero principal maximal ideal.

Existence of non-tame automorphisms in positive characteristic.

Abstract

In a recent paper it has been established that over an Artinian ring R all two-dimensional polynomial automorphisms having Jacobian determinant one are tame if R is a Q-algebra. This is a generalization of the famous Jung-Van der Kulk Theorem, which deals with the case that R is a field (of any characteristic). Here we will show that for tameness over an Artinian ring, the Q-algebra assumption is really needed: we will give, for local Artinian rings with square-zero principal maximal ideal, a complete description of the tame automorphism subgroup. This will lead to an example of a non-tame automorphism, for any characteristic p>0.