Extensions of the Frobenius to the ring of differential operators on a polynomial algebra in prime characteristic.

Let $ K$. It is proved that each automorphism $ \sigma \in \operatorname{Aut}_K(\mathcal D(P_n))$ $ \mathcal D(P_n)$ $ P_n= K[x_1, \ldots, x_n]$uniquely determined by the elements $ \sigma (x_1), \ldots ,\sigma (x_n)$ $ \operatorname{Frob}(\mathcal D(P_n))$ $ \mathcal D(P_n)$, to the ring $ \mathcal D(P_n)$ $ \operatorname{Aut}_K(\mathcal D(P_n) ) \cdot \mathcal{F}$ $ \mathcal{F}$ to $ \mathcal D(P_n)$ $ \mathcal{F}$countably many independent parameters). [ABSTRACT FROM AUTHOR]