31,017 research outputs found

### 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

### Weak length induction and slow growing depth boolean circuits

We define a hierarchy of circuit complexity classes LD^i, whose depth are the
inverse of a function in Ackermann hierarchy. Then we introduce extremely weak
versions of length induction and construct a bounded arithmetic theory L^i_2
whose provably total functions exactly correspond to functions computable by
LD^i circuits. Finally, we prove a non-conservation result between L^i_2 and a
weaker theory AC^0CA which corresponds to the class AC^0. Our proof utilizes
KPT witnessing theorem

### 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

### 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

### The GIT moduli of semistable pairs consisting of a cubic curve and a line on ${\mathbb P}^{2}$

We discuss the GIT moduli of semistable pairs consisting of a cubic curve and
a line on the projective plane. We study in some detail this moduli and compare
it with another moduli suggested by Alexeev. It is the moduli of pairs (with no
specified semi-abelian action) consisting of a cubic curve with at worst nodal
singularities and a line which does not pass through singular points of the
cubic curve. Meanwhile, we make a comparison between Nakamura's
compactification of the moduli of level three elliptic curves and these two
moduli spaces

### The Mosco convergence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains

We consider the convergent problems of Dirichlet forms associated with the
Laplace operators on thin domains. This problem appears in the field of quantum
waveguides. We study that a sequence of Dirichlet forms approximating the
Laplace operators with the delta potential on thin domains Mosco converges to
the form associated with the Laplace operator with the delta potential on the
graph in the sense of Gromov-Hausdorff topology. From this results we can make
use of many results established by Kuwae and Shioya about the convergence of
the semigroups and resolvents generated by the infinitesimal generators
associated with the Dirichlet forms.Comment: 16 pages, 5 figure

### Developing Takeuti-Yasumoto forcing

In late 90's G.Takeuti and Y.Yasumoto gave forcing constructions for bounded
arithmetic. We will reformulate their constructions using two-sort bounded
arithmetic and prove the followings. 1. Generic extensions are related with
P=NP problem. 2. J.Krajicek's forcing constructions can be given as
Takeuti-Yasumoto forcing. 3. We can either satisfy or falsify the dual weak
pigeonhole principles in generic extensions.Comment: 17 page

### 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

### 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 $[k({\bf x}):L]=2$

- β¦