On polynomial functions from \(\mathbb{Z}_ n\) to \(\mathbb{Z}_ m\)
type: Item
instance of: scholarly article
title: On polynomial functions from \(\mathbb{Z}_ n\) to \(\mathbb{Z}_ m\)
zbMATH Open document ID: 0849.11028
DOI: 10.1016/0012-365X(93)E0162-W
author: Zhibo Chen
published in: Discrete Mathematics
publication date: 1996-11-12T00:00:00Z
review text: Let \(\mathbb{Z}_n\) denote the ring of residues modulo \(n\). The author calls a function \(f: \mathbb{Z}_n\to \mathbb{Z}_m\) a polynomial function if there is a polynomial \(F\in \mathbb{Z} [x ]\) such that \(f(j) \equiv F(j) \pmod m\) for \(j=1, \dots, m-1\). (It should be observed that a different choice of representatives for \(\mathbb{Z}_n\) would lead to a different class.) With \(\lambda\) being the smallest integer such that \(m|\lambda!\) and \(\mu= \min(n, \lambda)\), he proves that these functions have a unique representation in the form \(\sum^{ \mu-1 }_{ k=0} a_k x^k \pmod m\) with \(0\leq a_k< m/ (m,k!)\). The total number of polynomial functions is determined and criteria are given for a function to be polynomial. It is shown that those pairs \((m,n)\) for which all functions are polynomial are exactly those for which \(n\) is not greater than the least prime factor of \(m\).
Mathematics Subject Classification ID: 11B83
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11B34
zbMATH DE Number: 719455
zbMATH Keywords: residue classes
zbMATH Keywords: polynomial functions
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 719455