  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.
