A generalization of Nakai's theorem on locally finite iterative higher derivations

Let $k$ be a field of arbitrary characteristic. Nakai (1978) proved a structure theorem for $k$-domains admitting a nontrivial locally finite iterative higher derivation when $k$ is algebraically closed. In this paper, we generalize Nakai’s theorem to cover the case where $k$ is not algebraically closed. As a consequence, we obtain a cancellation theorem of the following form: Let $A$ and $A^{\mathrm{\prime }}$ be finitely generated $k$-domains with $A[x]\simeq _kA’[x]$\mathrm{.}If$A$and$\bar{k}\otimes _kA$areUFDsand$\mathop{\rm trans.deg}\nolimits_kA=2$,thenwehave$A\simeq _kA’$. This generalizes the cancellation theorem of Crachiola (2009).