What's new about integer-valued polynomials on a subset? (Q2752903)

If \(E\) is a subset of the quotient field \(K\) of a domain \(D\), the set \(\text{Int}(E,D)\) consisting of univariate polynomials \(f\in K[X]\) with \(f(E)\subseteq D\) is the ring of integer-valued polynomials on \(E\). The paper under review is a survey of recent results referring to infinite subsets \(E\) of the quotient field of a rank-one valuation domain \(V\). The standing hypotheses are: NEWLINENEWLINENEWLINE\(dE\subseteq V\) for a nonzero element \(d\) of \(V\) and the completion \(\widehat{E}\) of \(E\) in the radical topology of \(V\) is compact. NEWLINENEWLINENEWLINEIn this framework, the Stone-Weierstrass theorem does hold: NEWLINENEWLINENEWLINE\(\text{Int}(E,V)\) is dense (in the uniform convergence topology) in the ring \({\mathcal C}(\widehat{E},\widehat{V})\) of continuous functions. NEWLINENEWLINENEWLINEConsequently, one may describe the prime ideals in \(\text{Int}(E,V)\). One can also show that \(\text{Int}(E,V)\) is a Prüfer domain whose finitely generated ideals are characterized by the values of the polynomials they contain. An entire section is devoted to the study of the \(V\)-module structure on \(\text{Int}(E,V)\). Finally one mentions extensions of these results to integer-valued multivariate polynomials.NEWLINENEWLINEFor the entire collection see [Zbl 0964.00012].