Factorial preservation
Let \(A\) be an integral domain, \(S\) a subset of \(A\), and \(\text{Int}(S,A)\) the \(A\)-module of the integer-valued polynomials over \(S\). \textit{M. Bhargava} [J. Number Theory 72, No. 1, 67--75 (1998; Zbl 0931.13004)] gives the following definition: For \(n\in \mathbb N\), the \(n\)-th factorial of \(S\) relative to \(A\) is the ideal of \(A\) defined by \((n!)_S^A=\{y\in A\mid yP(X)\in A[X] \, \text{for all}\;P(X)\in\text{Int}(S,A)\;\text{with deg}(P(X))\leq n\}\). The main result of this paper is the following. Let \(A\) be a Dedekind domain with finite residue fields and with finite group of units \(U(A)\). Let \(S\) be an infinite subset of \(A\) and \(f(X)\in\text{Int}(S,A)\). Then \((n!)_S^A=(n!)_{f(S)}^A\) for all \(n\in\mathbb N\) if and only if \(f(X)=uX+a\), where \(u\in U(A)\) and \(a\in A\). In particular, the author gives an answer to the question addressed by \textit{R. Gilmer} and \textit{W. W. Smith} [Arch. Math. 73, No. 5, 355--365 (1999; Zbl 0955.13009)] about \(f(X)\) when \(A=\mathbb Z\) and \(\text{Int}(S,A)=\text{Int}(f(S),A)\).
scientific article; zbMATH DE number 2130605