Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism

TL;DR

This paper revisits the Shestakov-Umirbaev theory on polynomial automorphisms, utilizing a generalized inequality to refine tameness criteria and confirm that certain automorphisms cannot be tame.

Contribution

It reconstructs the Shestakov-Umirbaev theory using a new generalized inequality, providing a more precise criterion for polynomial automorphism tameness.

Findings

No tame automorphism admits a reduction of type IV

Reconstruction of the theory with a generalized inequality

Enhanced criteria for polynomial automorphism tameness

Abstract

In 2003, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the "generalized Shestakov-Umirbaev inequality", which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we show that no tame automorphism of a polynomial ring admits a reduction of type IV.