Characterization of complete exterior sets of conics
type: Item
instance of: scholarly article
title: Characterization of complete exterior sets of conics
zbMATH Open document ID: 0761.51006
DOI: 10.1007/BF01204717
author: H. A. Wilbrink
author: Aart Blokhuis
author: Ákos Seress
published in: Combinatorica
publication date: 1993-01-16T00:00:00Z
review text: Let \(q\) be an odd prime power, and let \(\pi=PG(2,q)\) denote the desarguesian projective plane of order \(q\). A complete exterior set is defined to be a set of \((q+1)/2\) exterior points with respect to a fixed nondegenerate conic of \(\pi\) such that any two of these points determine a passant line. (Of course, the maximum size of any set of such points is \((q+1)/2.)\) Clearly, the \((q+1)/2\) exterior points on any passant line form a complete exterior set. In the paper under review the authors show that these are the only possible examples when \(q\equiv 1\pmod 4\). For \(q=7\), 11, 19, 23, 27 and 31 nonlinear examples of complete exterior sets are found (exhaustively by computer). However, it is conjectured that for \(q>31\) the only examples of complete exterior sets are the linear ones. The main idea behind the proof in the \(q\equiv 1\pmod 4\) case is to carefully coordinatize so that the problem may be reformulated as follows: characterize all bijections \(\Phi:GF(q)^*\to GF(q)^*\) of order 2, sending squares to nonsquares, such that for any two squares \(s\) and \(t\) we have \((s-t)(s-\Phi(t))(\Phi(s)-t)(\Phi(s)-\Phi(t))\) is a nonsquare. The bijections of the form \(\Phi(x)=n/x\) for some fixed nonsquare \(n\) in \(GF(q)\) correspond to the linear complete exterior sets. To show that these are the only possible bijections when \(q\equiv 1\pmod 4\) the authors cleverly use another reformulation of the problem in terms of the Paley graph as well as the following consequence of a theorem of \textit{L. Carlitz} and \textit{R. McConnel} ['A theorem on permutations in a finite field', Proc. Am.Math. Soc. 11, 456-459 (1960)]: Let \(f:GF(q)\to GF(q)\) be a function with the property that for all \(x,y\in GF(q)\) with \(x\neq y\), \((f(x)-f(y))/(x-y)\) is a nonsquare. Then \(f\) is of the form \(f(x)=a+bx^{p^ j}\), where \(a,b\in GF(q)\) with \(b\) a nonsquare and \(p\) is the characteristic of the field.
Mathematics Subject Classification ID: 51E15
zbMATH DE Number: 89339
zbMATH Keywords: nondegenerate conic
zbMATH Keywords: exterior points
zbMATH Keywords: passants
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
cites work: Q3982810
cites work: Inversive geometry and some new translation planes. I
cites work: A Theorem on Permutations of a Finite Field
cites work: A Theorem on Permutations in a Finite Field
cites work: Example of a chain of circles on an elliptic quadric of PG(3,q),q=7,11
cites work: Pseudo-ordered polynomials over a finite field
full work available at URL: https://doi.org/10.1007/bf01204717
OpenAlex ID: W2001379196
description: scientific article; zbMATH DE number 89339