Complete arcs containing \((q+1)/2\) points of a conic in Galois fields of odd order
type: Item
instance of: scholarly article
title: Complete arcs containing \((q+1)/2\) points of a conic in Galois fields of odd order
zbMATH Open document ID: 0793.51005
DOI: 10.1007/BF02843949
author: V. Pereyra
publication date: 1994-01-26T00:00:00Z
review text: Given a \(k\)-arc \(H\) in a projective plane, a point of the plane is said to be \(H\)-free if no secant of \(H\) (chord of \(H\)) passes through the point. The author investigates the following problem. In a Desarguesian plane, \(PG(2,q)\), \(q\) odd, let \(\Gamma\) be a non-degenerate conic and let \(H\) be a set of \((q+1)/2\) points of \(\Gamma\) such that there exist at least two \(H\)-free interior points to \(\Gamma\) on an exterior line of \(\Gamma\). Determine the orders of complete arcs other than \(\Gamma\) which contain \(H\). Letting \(A\) and \(B\) denote the hypothesized points and \(O\) the pole (with respect to \(\Gamma\)) of the line containing them, the author shows that when \(q>13\) an \(H\) can be constructed so that, \(H\cup\{A,B\}\) is a complete arc, and if \(q=4t+1\), \(H\cup\{O\}\) is also a complete arc. The technique involves introducing coordinates drawn from the ring of residues modulo \((q+1)\), and using some number-theoretic properties of certain sum-free sets in that ring. This paper is a continuation of earlier work of the author and there is a heavy reliance on these previous papers.
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 490577
zbMATH Keywords: complate arcs
zbMATH Keywords: conics
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
cites work: Q3851217
cites work: Le geometrie di Galois
cites work: Q3272923
cites work: Q5830713
full work available at URL: https://doi.org/10.1007/bf02843949
OpenAlex ID: W2335532532
published in: Rendiconti del Circolo Matematico di Palermo
Recommended article: Q3695927
Recommended article: Q3361483
Recommended article: Q4027808
Recommended article: Q3209737
Recommended article: Q5186258
Recommended article: Q4398591
Recommended article: Q3974659
Recommended article: Complete arcs and algebraic curves in \(\text{PG}(2,q)\)
Recommended article: Q3732449
Recommended article: Hasse-Weil theorem and construction of complete arcs of little cardinality in Galois planes of odd order
description: scientific article; zbMATH DE number 490577