On the Kara\'s type theorems for the multidegrees of polynomial automorphisms

TL;DR

This paper unifies various results on multidegrees of polynomial automorphisms in three variables by applying a generalized version of the Shestakov-Umirbaev theory, advancing understanding of tame automorphisms.

Contribution

It provides a strong unifying theorem that consolidates previous necessary conditions for multidegrees of tame automorphisms using an extended theory.

Findings

Unified several conditions into a comprehensive theorem

Extended Shestakov-Umirbaev theory to multidegree analysis

Enhanced criteria for identifying tame automorphisms

Abstract

To solve Nagata's conjecture, Shestakov-Umirbaev constructed a theory for deciding wildness of polynomial automorphisms in three variables. Recently, Kara\'s and others study multidegrees of polynomial automorphisms as an application of this theory. They give various necessary conditions for triples of positive integers to be multidegrees of tame automorphisms in three variables. In this paper, we prove a strong theorem unifying these results using the generalized Shestakov-Umirbaev theory.