On a generalization of Rédei's theorem
type: Item
instance of: scholarly article
title: On a generalization of Rédei's theorem
zbMATH Open document ID: 1050.51006
author: András Gács
published in: Combinatorica
publication date: 2004-10-19T00:00:00Z
review text: Let \(p\) be a prime and consider a set \(U\) of \(p\) points in AG\((2,p)\). Let \(N\) be the number of distinct ``directions'' or slopes (including \(\infty\)) of all lines resulting from connecting any two points in \(U\). The case \(p=2\) is trivial, and the case \(p=3\) is easy, so we can assume \(p\geq5\). Then the following cases occur (i) \(U\) is a line; (ii) \(N={p+3\over 2}\) and \(U\) is an example given by \textit{L. Lovász} and \textit{A. Schrijver} [Stud. Sci. Math. Hung. 16, 449--454 (1981; Zbl 0535.51009)]; (iii) \(N\geq\lfloor2{p-1\over 3}\rfloor+1\) (the author's main result). A simple example with \(N=2{p-1\over 3}+2\) for \(p\equiv1\bmod3\) is given. This shows that the bound is not far from being sharp. Some remarks are made for the case when \(p\) is a prime power, rather than just a prime. For the proof the statement is translated into an algebraic setting, using an interpolation polynomial over the field with \(p\) elements. Another reformulation uses permutation polynomials.
reviewed by: Hubert Kiechle
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 2108655
zbMATH Keywords: Rédei's theorem
zbMATH Keywords: directions
MaRDI profile type: Publication
full work available at URL: s00493-003-0035-y
OpenAlex ID: W137889574
DOI: 10.1007/S00493-003-0035-Y
description: scientific article; zbMATH DE number 2108655