Polynomial automorphisms of characteristic order and their invariant rings

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.