  Let $K$ be a field of characteristic $p>0$. It is proved that each
automorphism $\s \in \Aut_K(\CDPn)$ of the ring $\CDPn$ of differential
operators on a polynomial algebra $P_n= K[x_1, ..., x_n]$ is {\em uniquely}
determined by the elements $\s (x_1), ... ,\s (x_n)$, and the set $\Frob
(\CDPn)$ of all the extensions of the Frobenius from certain maximal
commutative polynomial subalgebras of $\CDPn$, like $P_n$, is equal to
$\Aut_K(\CDPn) \cdot \CF$ where $\CF$ is the set of all the extensions of the
Frobenius from $P_n$ to $\CDPn$ that leave invariant the subalgebra of scalar
differential operators. The set
  $\CF$ is found explicitly, it is large (a typical extension depends on {\em
countably} many independent parameters).
