论文标题
特征顺序及其不变环的多项式自动形态
Polynomial automorphisms of characteristic order and their invariant rings
论文作者
论文摘要
令$ k $为特征$ p> 0 $的字段。我们讨论多项式环$ k [x_1，\ ldots，x_n] $的自动形态，或等效地，$ {\ bf z}/p {\ bf z} $ - 仿射空间$ {\ bf a} _k _k^n $上的$ {\ bf z}/p {\ bf z} $。当$ n = 2 $时，这种自动形态被认为是固定变量的自动形态的共轭。当$ n \ ge 3 $时，这是一个悬而未决的问题。 在本文中， （1）当$ n = 3 $时，我们对此问题进行了第一个反例。实际上，我们表明，每一个$ {\ bf g} _a $ - $ {\ bf a} _k^3 $等级的三个$ n = 3 $的反例。我们通过在$ {\ bf a} _k^3 $上构建一个排名第三$ {\ bf g} _a $ actions的家族来提供一个反例。 （2）对于这个由$ {\ bf g} _a $ - actions的家族引起的自动形态，我们表明不变的环在$ k [x_1，x_2，x_3] $ and Plinth Plinth Ideal是Plinth Idexal是主要的假设下，在某些温和的假设下，仅在$ k [x_1，x_2，x_3] $时。 （3）我们研究$ r [x_1，x_2] $的长田类型自动形态，其中$ r $是特征的UFD $ p> 0 $。这种类型的自动形态为$ p $。我们给出了不变环的必要条件，使得$ r [x_1，x_2] $是同构。这种情况等效于底座理想是主要的条件。
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.
