Rings of integer-valued polynomials and derivatives on finite sets (Q1945795)

Let \(D\) be an integral domain, \(K\) its field of fractions and \(E\) a finite subset of \(D\). For \(r=0,1,\dots\) denote by \(\mathrm{Int}^{(r)}(E,D)\) the set of polynomials \(f\in K[X]\) satisfying \(f^{(k)}(E)\subset K\) for \(k=0,1,\dots,r\), where \(f^{(0)}=f\) and for \(k\geq1\) \(f^{(k)}\) is the \(k\)-th derivative of \(f\). The authors prove (Theorem 3.3) that if \(D\) has the \(m\)-generator property, and either \(r\geq1\) or \(m\geq2\), then \(\mathrm{Int}^{(r)}(E,D)\) has the \((r+1)m\)-generator property. In a previous paper [Arch. Math., 86, No. 1, 36--42 (2006; Zbl 1093.13017)] the first author and \textit{J. Boynton} obtained this in the case \(r=0\) and proved also the converse implication. In Theorem 3.6. the authors show by an example that their result is best possible.