Nakayama automorphisms of Ore extensions over polynomial algebras

Nakayama automorphisms play an important role in several mathematical branches, which are known to be tough to compute in general. We compute the Nakayama automorphism $\nu$ of any Ore extension $R[x; \sigma, \delta]$ over a polynomial algebra $R$ in $n$ variables for an arbitrary $n$. The formula of $\nu$ is obtained explicitly. When $\sigma$ is not the identity map, the invariant $E^G$ is also investigated in term of Zhang's twist, where $G$ is a cyclic group sharing the same order with $\sigma$.