Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic
Abstract
Let K be a field of characteristic p>0. It is proved that each automorphism ∈ K() of the ring of differential operators on a polynomial algebra Pn= K[x1, ..., xn] is uniquely determined by the elements (x1), ... , (xn), and the set () of all the extensions of the Frobenius from certain maximal commutative polynomial subalgebras of , like Pn, is equal to K() · where is the set of all the extensions of the Frobenius from Pn to that leave invariant the subalgebra of scalar differential operators. The set is found explicitly, it is large (a typical extension depends on countably many independent parameters).
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