Factorial preservation
type: Item
instance of: scholarly article
title: Factorial preservation
zbMATH Open document ID: 1073.13011
author: Youssef Fares
published in: Archiv der Mathematik
publication date: 2005-01-20T00:00:00Z
review text: 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 \(\text{Int}(S,A)=\text{Int}(f(S),A)\).
reviewed by: Martine Picavet-L'Hermitte
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 05A10
zbMATH DE Number: 2130605
zbMATH Keywords: Integer-valued polynomial
zbMATH Keywords: \(n\)-th factorial of a subset
zbMATH Keywords: Dedekind domain
MaRDI profile type: Publication
full work available at URL: s00013-004-1089-9
OpenAlex ID: W4235122692
DOI: 10.1007/S00013-004-1089-9
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description: scientific article; zbMATH DE number 2130605