# Extensions of the Frobenius to ring of differential operators on polynomial algebra in prime characteristic - Mathematics > Rings and Algebras

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Abstract: Let $K$ be a field of characteristic $p>0$. It is proved that eachautomorphism $\s \in \Aut K\CDPn$ of the ring $\CDPn$ of differentialoperators on a polynomial algebra $P n= Kx 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 maximalcommutative 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 theFrobenius from $P n$ to $\CDPn$ that leave invariant the subalgebra of scalardifferential operators. The set$\CF$ is found explicitly, it is large a typical extension depends on {\emcountably} many independent parameters.

Author: ** V. V. Bavula**

Source: https://arxiv.org/