A divided-difference characterization of polynomials over a general field
type: Item
instance of: scholarly article
title: A divided-difference characterization of polynomials over a general field
zbMATH Open document ID: 0892.39020
DOI: 10.1007/PL00000046
author: Roy O. Davies
author: George Rousseau
published in: Aequationes Mathematicae
publication date: 1998-07-19T00:00:00Z
review text: Generalizing the characterization of real quadratic functions \(f\) by the property \((f(x_{1})-f(x_{2}))/\) \((x_{1}-x_{2})=h(x_{1}+x_{2})\), this reviewer has proved [Math. Mag. 58, 42-45 (1985; Zbl 0571.39005)] that this equation characterizes the polynomials \(f\) of second degree also among functions mapping an arbitrary field of characteristic different from 2 into itself. The generalization that polynomials of \(n\)th degree are characterized by the property that the \((n-1)\)st divided difference \(f[x_{1},\dots,x_{n}]\) depends only upon \(x_{1}+\dots+x_{n}\) has been proved by \textit{J. Schwaiger} [Aequationes Math. 48, No. 2-3, 317-323 (1994; Zbl 0810.39007)] in the case where the number of elements of the field is at least \(8n-14\). The present authors eliminate this restriction. Furthermore, they consider the case of characteristic 2, pointing out also an error in a counter-example in the reviewer's paper quoted above.
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 1130461
zbMATH Keywords: functional equations
zbMATH Keywords: polynomials
zbMATH Keywords: divided differences
zbMATH Keywords: fields
zbMATH Keywords: characteristic
zbMATH Keywords: quadratic functions
reviewed by: János Aczél
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1130461