Summary: We present some new results on and approaches to integer-valued polynomial rings. One of our results is that, for any PvMD \(D\), the domain Int\((D)\) of integer-valued polynomials on \(D\) is locally free as a \(D\)-module if Int\((D_{\mathfrak p})=\text{Int}(D))_{\mathfrak p}\) for every prime ideal \({\mathfrak p}\) of \(D\). This fact allows us in particular to strengthen the main results of [J. Algebra 318, 68–92 (2007; Zbl 1129.13022)], to prove, for example, that the multivariable integer-valued polynomial ring Int\((D^n)\) decomposes as the \(n\)-th tensor power of Int\((D)\) over \(D\) for any such PuMD \(D\). We also present a survey of some new techniques for studying integer-valued polynomial rings – such as universal properties, tensor product decompositions, pullback constructions, and Bhargava rings – that may prove useful.
For the entire collection see [Zbl 1175.13001].