# zbMATH — the first resource for mathematics

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
##### Citations:
Zbl 0955.13009; Zbl 0931.13004
Full Text: