Co-tame polynomial automorphisms
Eric Edo, Drew Lewis

TL;DR
This paper introduces new classes of co-tame polynomial automorphisms, especially in three dimensions, and establishes conditions under which automorphisms are co-tame, using a novel class of maps called 'translation degenerate automorphisms.'
Contribution
The paper proves that certain classes of automorphisms, including 3-triangular automorphisms, are co-tame, and characterizes when m-triangular automorphisms are either affine or co-tame in three dimensions.
Findings
3-triangular automorphisms are co-tame.
The statement 'every m-triangular automorphism is affine or co-tame' holds iff m ≤ 3 in dimension 3.
All translation degenerate automorphisms are co-tame.
Abstract
A polynomial automorphism of 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, including -triangular automorphisms, are co-tame. Of particular interest, if , we show that the statement "Every -triangular automorphism is either affine or co-tame" is true if and only if ; this improves upon positive results of Bodnarchuk (for , in any dimension ) and negative results of the authors (for , ). The main technical tool we introduce is a class of maps we term 'translation degenerate automorphisms'; we show that all of these are co-tame, a result that may be of independent interest in the further study of co-tame automorphisms.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Mathematical Dynamics and Fractals · Quantum chaos and dynamical systems
