Positive characteristic analogs of closed polynomials (Q632291)

Let \(k\) be a field of zero characteristics. A non-constant polynomial \(f\in k[X_1,\dots,X_n]\) is said to be closed if the ring \(k[f]\) is integrally closed in \(k[X_1,\dots,X_n]\). A characterization of closed polynomials in terms of derivations was for \(n=2\) given by \textit{A. Nowicki} [Nagoya Math. J., 109, 151--157 (1988; Zbl 0642.13016)] who showed that \(f\) is closed if and only if \(k[f]\) is the ring of constants of a certain \(k\)-derivation. The author observes that for fields of positive characteristics this assertion fails, and discusses the possibilities of its modification in that case.