Closed polynomials in polynomial rings over integral domains
A polynomial \(f(X_1,\dots,X_n)\) over an integral domain \(R\) is called a closed polynomial, if the ring \(R[f]\) is integrally closed in \(R[X_1,\dots,X_n]\). These polynomials have been studied earlier in the case when \(R\) is a field (see e.g. [\textit{A. Nowicki}, Nagoya Math. J. 109, 151--157 (1988; Zbl 0642.13016); \textit{M. Ayad}, Acta Arith. 105, No. 1, 9--28 (2002; Zbl 1038.12001); \textit{I. V. Arzhantsev} and \textit{A. P. Petravchuk}, Ukr. Mat. Zh. 59, No. 12, 1587--1593 (2007; Zbl 1164.13302)]), as well as in the case when \(R\) is a unique factorization domain [\textit{M. Kato} and \textit{H. Kojima}, Commun. Algebra 43, No. 5, 1935--1938 (2015; Zbl 1327.13033)]. The authors treat now the case when \(R\) is an arbitrary integral domain, and generalize the results of papers quoted above, establishing in particular characterizations of closed polynomials. We quote one typical result (second part of Theorem 3.1): If \(R\) is an integral domain with quotient field \(K\) of zero characteristic, and the polynomial \(f\in R[X_1,\dots,X_n]\setminus R\) satisfies \(K[f]\cap R[X_1,\dots,X_n] = R[f]\), then \(f\) is closed if and only if there exists an \(R\)-derivation \(d\) on \(R[X_1,\dots,X_n]\) with Ker \(d = R[f]\).
scientific article; zbMATH DE number 6476979