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Let $X$ be an integral affine scheme of characteristic $p>0$, and $\sigma $ a non-identity automorphism of $X$. If $\sigma $ is $\textit{exponential}$, i.e., induced from a ${\bf G}_a$-action on $X$, then $\sigma $ is obviously of order $p$. It is easy to see that the converse is not true in general. In fact, there exists $X$ which admits an automorphism of order $p$, but admits no non-trivial ${\bf G}_a$-actions. However, the situation is not clear in the case where $X$ is the affine space ${\bf A}_R^n$, because ${\bf A}_R^n$ admits various ${\bf G}_a$-actions as well as automorphisms of order $p$. In this paper, we study exponentiality of automorphisms of ${\bf A}_R^n$ of order $p$, where the difficulty stems from the non-uniqueness of ${\bf G}_a$-actions inducing an exponential automorphism. Our main results are as follows. (1) We show that the triangular automorphisms of ${\bf A}_R^n$ of order $p$ are exponential in some low-dimensional cases. (2) We construct a non-exponential automorphism of ${\bf A}_R^n$ of order $p$ for each $n\ge 2$. Here, $R$ is any UFD which is not a field. (3) We investigate the ${\bf G}_a$-actions inducing an elementary automorphism of ${\bf A}_R^n$.

Let $k$ be a field of characteristic $p>0$. We discuss the automorphisms of the polynomial ring $k[x_1,\ldots ,x_n]$ of order $p$, or equivalently the ${\bf Z}/p{\bf Z}$-actions on the affine space ${\bf A}_k^n$. When $n=2$, such an automorphism is know to be a conjugate of an automorphism fixing a variable. It is an open question whether the same holds when $n\ge 3$. In this paper, (1) we give the first counterexample to this question when $n=3$. In fact, we show that every ${\bf G}_a$-action on ${\bf A}_k^3$ of rank three yields counterexamples for $n=3$. We give a family of counterexamples by constructing a family of rank three ${\bf G}_a$-actions on ${\bf A}_k^3$. (2) For the automorphisms induced by this family of ${\bf G}_a$-actions, we show that the invariant ring is isomorphic to $k[x_1,x_2,x_3]$ if and only if the plinth ideal is principal, under some mild assumptions. (3) We study the Nagata type automorphisms of $R[x_1,x_2]$, where $R$ is a UFD of characteristic $p>0$. This type of automorphisms are of order $p$. We give a necessary and sufficient condition for the invariant ring to be isomorphic to $R[x_1,x_2]$. This condition is equivalent to the condition that the plinth ideal is principal.

The notion of initial ideal for an ideal of a polynomial ring appears in the theory of Gr\"obner basis. Similarly to the initial ideals, we can define the initial algebra for a subalgebra of a polynomial ring, or more generally of a Laurent polynomial ring, which is used in the theory of SAGBI (Subalgebra Analogue to Gr\"obner Bases for Ideals) basis. The initial algebra of a finitely generated subalgebra is not always finitely generated, and no general criterion for finite generation is known. The aim of this paper is to present a new class of finitely generated subalgebras having non-finitely generated initial algebras. The class contains a subalgebra for which the set of initial algebras is continuum, as well as a subalgebra with finitely many distinct initial algebras.

In this article, we give a new class of automorphisms of Leavitt path algebras of arbitrary graphs. Consequently, we obtain Anick type automorphisms of these Leavitt path algebras and new irreducible representations of Leavitt algebras of type $(1, n)$., Comment: 20 pages

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$., Comment: 14 pages, to appear in Transformation Groups

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

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.

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.

For a field $k$, the ring of invariants of an action of the unipotent $k$-group $G_a$ on an affine $k$-variety is quasi-affine, but not generally affine. Cable algebras are introduced as a framework for studying these invariant rings. It is shown that the ring of invariants for the $G_a$-action on $A^5_k$ constructed by Daigle and Freudenburg is a monogenetic cable algebra. A generating cable is constructed for this ring, and a complete set of relations is given as a prime ideal in the infinite polynomial ring over $k$. In addition, it is shown that the ring of invariants for the well-known $G_a$-action on $A^7_k$ due to Roberts is a cable algebra., Comment: 29 pages

Let $k$ be a field of arbitrary characteristic. Nakai (1978) proved a structure theorem for $k$-domains admitting a nontrivial locally finite iterative higher derivation when $k$ is algebraically closed. In this paper, we generalize Nakai's theorem to cover the case where $k$ is not algebraically closed. As a consequence, we obtain a cancellation theorem of the following form: Let $A$ and $A'$ be finitely generated $k$-domains with $A[x]\simeq _kA'[x]$. If $A$ and $\bar{k}\otimes _kA$ are UFDs and $\mathop{\rm trans.deg}\nolimits_kA=2$, then we have $A\simeq _kA'$. This generalizes the cancellation theorem of Crachiola (2009).

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.

Let $R$ be an integral domain over a field $k$, and $G$ a subgroup of the automorphism group of the polynomial ring $R[x_1,..., x_n]$ over $R$. In this paper, we discuss when $G$ is diagonalizable under the assumption that $G$ is diagonalizable over the field of fractions of $R$. We are particularly interested in the case where $G$ is a finite abelian group. Kraft-Russell (2014) implies that every finite abelian subgroup of ${\rm Aut}_R(R[x_1,x_2])$ is diagonalizable if $R$ is an affine PID over $k={\bf C}$. One of the main results of this paper says that the same holds for a PID $R$ over any field $k$ containing enough roots of unity.

In this brief overview we will reflect on the process of the design of the linear collider (LC) final focus (FF) optics, and will also describe the theoretical and experimental efforts on design and practical realisation of a prototype of the LC FF optics implemented in the ATF2 facility at KEK, Japan, presently being commissioned and operated.

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 pages

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.

The Derksen--Hadas--Makar-Limanov theorem (2001) says that the invariants for nontrivial actions of the additive group on a polynomial ring have no intruder. In this paper, we generalize this theorem to the case of stable invariants.

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.

We study wildness of automorphisms of a polynomial ring in three variables in detail using the Shestakov-Umirbaev theory and its generalization., Comment: 168 pages

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 pages

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 answered

We give a generalization of the Shestakov-Umirbaev inequality which plays an important role in their solution of the tame generators conjecture.