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

Stably co-tame polynomial automorphisms over commutative rings

We say that a polynomial automorphism $\phi$ in $n$ variables is stably co-tame if the tame subgroup in $n$ variables is contained in the subgroup generated by $\phi$ and affine automorphisms in $n+1$ variables. In this paper, we give conditions for stably co-tameness of polynomial automorphisms

Degeneration of tame automorphisms of a polynomial ring

Recently, Edo-Poloni constructed a family of tame automorphisms of a polynomial ring in three variables which degenerates to a wild automorphism. In this note, we generalize the example by a different method

The automorphism theorem and additive group actions on the affine plane

Due to Rentschler, Miyanishi and Kojima, the invariant ring for a ${\bf G}_a$-action on the affine plane over an arbitrary field is generated by one coordinate. In this note, we give a new short proof for this result using the automorphism theorem of Jung and van der Kulk

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.Comment: 52 page

Hilbert's fourteenth problem and field modifications

Let $k({\bf x})=k(x_1,\ldots ,x_n)$ be the rational function field, and $k\subsetneqq L\subsetneqq k({\bf x})$ an intermediate field. Then, Hilbert's fourteenth problem asks whether the $k$-algebra $A:=L\cap k[x_1,\ldots ,x_n]$ is finitely generated. Various counterexamples to this problem were already given, but the case $[k({\bf x}):L]=2$ was open when $n=3$. In this paper, we study the problem in terms of the field-theoretic properties of $L$. We say that $L$ is minimal if the transcendence degree $r$ of $L$ over $k$ is equal to that of $A$. We show that, if $r\ge 2$ and $L$ is minimal, then there exists $\sigma \in {\mathop{\rm Aut}\nolimits}_kk(x_1,\ldots ,x_{n+1})$ for which $\sigma (L(x_{n+1}))$ is minimal and a counterexample to the problem. Our result implies the existence of interesting new counterexamples including one with $n=3$ and $[k({\bf x}):L]=2$

Weighted multidegrees of polynomial automorphisms over a domain

The notion of the weighted degree of a polynomial is a basic tool in Affine Algebraic Geometry. In this paper, we study the properties of the weighted multidegrees of polynomial automorphisms by a new approach which focuses on stable coordinates. We also present some applications of the generalized Shestakov-Umirbaev theory

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

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.Comment: Question 4.1 of the first version was answere

Generalisations of the tame automorphisms over a domain of positive characteristic

In this paper, we introduce two generalizations of the tame subgroup of the automorphism group of a polynomial ring over a domain of positive characteristic. We study detailed structures of these new `tame subgroups' in the case of two variables.Comment: 20 page