Congruence preservation and polynomial functions from \(\mathbb{Z}_n\) to \(\mathbb{Z}_m\)
type: Item
instance of: scholarly article
title: Congruence preservation and polynomial functions from \(\mathbb{Z}_n\) to \(\mathbb{Z}_m\)
zbMATH Open document ID: 0883.11011
DOI: 10.1016/S0012-365X(96)00093-3
author: Manjul Bhargava
published in: Discrete Mathematics
publication date: 1998-02-02T00:00:00Z
review text: A function from \(\{0,1,\dots,n-1\}\) to \(\mathbb{Z}_m\) is called congruence-preserving if \(a\equiv b\pmod d\) implies \(f(a)\equiv f(b)\pmod d\) for all \(d\mid m\). The number of congruence-preserving functions is evaluated. A comparison with a formula of \textit{Z. Chen} [Discrete Math. 137, 137--145 (1995; Zbl 0849.11028)] for the number of polynomials modulo \(m\) yields a sufficient and necessary condition for all congruence-preserving functions to be polynomials. In particular, if \(m=n\) then this happens if and only if \(m\) is not divisible by 8, nor by the square of any odd prime. The density of those numbers is calculated (\(=7/\pi^2\)).
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11A07
Mathematics Subject Classification ID: 11C08
zbMATH DE Number: 1062436
zbMATH Keywords: congruences
zbMATH Keywords: polynomials
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
cites work: On polynomial functions from \(\mathbb{Z}_ n\) to \(\mathbb{Z}_ m\)
full work available at URL: s0012-365x(96)00093-3
OpenAlex ID: W2100617828
description: scientific article; zbMATH DE number 1062436