243 research outputs found

### 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

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