Wildness of polynomial automorphisms in three variables
We study wildness of automorphisms of a polynomial ring in three variables in detail using the Shestakov-Umirbaev theory and its generalization.
math.AC↗
arXiv subjects
Shigeru Kuroda
Publications and source records attributed to Shigeru Kuroda.
Shestakov-Umirbaev reductions and Nagata's conjecture on a polynomial automorphism
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.
math.AC↗
Automorphisms of a polynomial ring which admit reductions of type I
Recently, Shestakov-Umirbaev solved Nagata's conjecture on an automorphism of a polynomial ring. To solve the conjecture, they defined notions called reductions of types I--IV for automorphisms of a polynomial ring. An automorphism admitting a reduction of type I was first found by Shestakov-Umirbaev. Using a computer, van den Essen--Makar-Limanov--Willems gave a family of such automorphisms. In this paper, we present a new construction of such automorphisms using locally nilpotent derivations. As a consequence, we discover that there exists an automorphism admitting a reduction of type I which satisfies some degree condition for each possible value.
math.AC↗
A generalization of the Shestakov-Umirbaev inequality
We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the tame generators conjecture.
math.AC↗