Skolem properties for several indeterminates (Q2785928)

Let \(D\) be an integral domain with quotient field \(K\), \(r\in\mathbb{N}\) and \(\text{Int} (D^r)\) the ring of \(D\)-valued polynomial functions on \(D^r\). The author investigates the so-called Skolem property for \(\text{Int} (D^r)\), which means that for each finitely generated ideal \(I\) of \(\text{Int} (D^r)\), such that \(I(a)=D\) for each \(a\in D^r\), the equality \(I= \text{Int} (D^r)\) holds -- here \(I(a)= \{f(a), f\in I\}\). The author gives a constructive proof of the already known result that \(\text{Int} (D^r)\) has that property if \(D\) is an order of a number field. Concerning the related almost strong Skolem property he derives a sufficient condition, which is fulfilled in particular for any ring of integers of a number field.NEWLINENEWLINEFor the entire collection see [Zbl 0855.00015].