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

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.