Maximal arcs in Desarguesian planes of odd order do not exist
type: Item
instance of: scholarly article
title: Maximal arcs in Desarguesian planes of odd order do not exist
zbMATH Open document ID: 0880.51003
DOI: 10.1007/BF01196129
published in: Combinatorica
publication date: 1998-01-05T00:00:00Z
review text: The title of the paper is exactly stating the contents of the paper. A maximal arc \({\mathcal K}\) of degree \(d\) in a projective plane of order \(n\) is a set of points such that each line of the plane intersects the set in 0 or \(d\) points (hence \(|{\mathcal K}|=nd-n+d\)). If \(1<d<n\), then all the known examples have \(d\) even (note that \(d\) has to divide \(n\)). It is a long standing conjecture that no maximal arcs of odd degree \(d\), \(1<d<n\), exist in projective planes of odd order \(n\). The authors prove that this conjecture is true in case of a Desarguesian plane, no use to say that this is an outstanding result. The proof (which is rather complicated) is based on the so-called polynomial technique. A more simplified proof by the first two authors is going to appear.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51A30
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1099679
zbMATH Keywords: maximal arcs
zbMATH Keywords: Desarguesian projective planes
author: Simeon Ball
author: Aart Blokhuis
author: Francesco Mazzocca
MaRDI profile type: Publication
cites work: Q3287732
cites work: Some maximal arcs in finite projective planes
cites work: Sets of type \((m,n)\) in the affine and projective planes of order nine
cites work: Construction of maximal arcs and partial geometries
cites work: Some results concerning \(\{(q+1)(n-1);\;n\}\)-arcs and \(\{(q+1)\;(n-1)+1;\;n\}\)-arcs in finite projective planes of order \(q\)
cites work: Construction of maximal arcs and dual ovals in translation planes
full work available at URL: https://doi.org/10.1007/bf01196129
OpenAlex ID: W2106257734
description: scientific article; zbMATH DE number 1099679