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Factorial preservation. (English) Zbl 1073.13011
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 \(S\).
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 R. Gilmer and W. W. Smith [Arch. Math. 73, No. 5, 355–365 (1999; Zbl 0955.13009)] about \(f(X)\) when \(A=\mathbb Z\) and \(\text{Int}(S,A)=\text{Int}(f(S),A)\).

MSC:
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
11B65 Binomial coefficients; factorials; \(q\)-identities
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
13B25 Polynomials over commutative rings
05A10 Factorials, binomial coefficients, combinatorial functions
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