The inversion formulae for automorphisms of polynomial algebras and differential operators in prime characteristic

Abstract

Let $K$ be an {\em arbitrary} field of characteristic $p>0$, let $A$ be one of the following algebras: $P_n:= K[x_1, ..., x_n]$ is a polynomial algebra, $\CD (P_n)$ is the ring of differential operators on $P_n$, $\CD (P_n)\t P_m$, the $n$&#39;th {\em Weyl} algebra $A_n$, the $n$&#39;th {\em Weyl} algebra $A_n\t P_m$ with polynomial coefficients $P_m$, the power series algebra $K[[x_1, ..., x_n]]$, $T_{k_1, ..., k_n}$ is the subalgebra of $\CD (P_n)$ generated by $P_n$ and the higher derivations $\der_i^{[j]}$, $0\leq j <p^{k_i}$, $i=1, ..., n$ (where $k_1, ..., k_n\in \mathbb{N}$), $T_{k_1, ..., k_n}\t P_m$, an {\em arbitrary central simple} (countably generated) algebra over an {\em arbitrary} field. {\em The inversion formula} for automorphisms of the algebra $A$ is found {\em explicitly}.