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

Let K be an arbitrary field of characteristic p>0. We find an explicit formula for the inverse of any algebra automorphism of any of the following algebras: the polynomial algebra Pncolon, equalsK[x1,…,xn], the ring of differential operators image on Pn, image, the n’th Weyl algebra An or Ancircle times operatorPm, the power series algebra K[[x1,…,xn]], the subalgebra Tk1,…,kn of image generated by Pn and the higher derivations image, 0≤j