Co-tame polynomial automorphisms

Abstract

A polynomial automorphism of $\mathbb{A}^n$ over a field of characteristic zero is called co-tame if, together with the affine subgroup, it generates the entire tame subgroup. We prove some new classes of automorphisms of $\mathbb{A}^n$, including nonaffine $3$-triangular automorphisms, are co-tame. Of particular interest, if $n=3$, we show that the statement “Every m-triangular automorphism is either affine or co-tame” is true if and only if $m=3$; this improves upon positive results of Bodnarchuk (for $m=2$, in any dimension $n$) and negative results of the authors (for $m=6$, $n=3$). The main technical tool we introduce is a class of maps we term translation degenerate automorphisms; we show that all of these are either affine or co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.

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