Inherited conics in Hall planes

Aart Blokhuis, István Kovács, Gábor P. Nagy, T. Szőnyi

Research output: Contribution to journalArticle

Abstract

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

Original languageEnglish
Pages (from-to)1098-1107
Number of pages10
JournalDiscrete Mathematics
Volume342
Issue number4
DOIs
Publication statusPublished - Apr 1 2019

Fingerprint

Arc of a curve
Hyperoval
Lemma
Triangle
Configuration

Keywords

  • Arcs
  • Finite projective planes
  • Hall planes

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

Cite this

Inherited conics in Hall planes. / Blokhuis, Aart; Kovács, István; Nagy, Gábor P.; Szőnyi, T.

In: Discrete Mathematics, Vol. 342, No. 4, 01.04.2019, p. 1098-1107.

Research output: Contribution to journalArticle

Blokhuis, A, Kovács, I, Nagy, GP & Szőnyi, T 2019, 'Inherited conics in Hall planes', Discrete Mathematics, vol. 342, no. 4, pp. 1098-1107. https://doi.org/10.1016/j.disc.2018.12.009
Blokhuis, Aart ; Kovács, István ; Nagy, Gábor P. ; Szőnyi, T. / Inherited conics in Hall planes. In: Discrete Mathematics. 2019 ; Vol. 342, No. 4. pp. 1098-1107.
@article{fbd7a376a91643bc9d05d4e1024a80e3,
title = "Inherited conics in Hall planes",
abstract = "The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchm{\'a}ros on Desargues configurations with perspective triangles inscribed in a conic.",
keywords = "Arcs, Finite projective planes, Hall planes",
author = "Aart Blokhuis and Istv{\'a}n Kov{\'a}cs and Nagy, {G{\'a}bor P.} and T. Szőnyi",
year = "2019",
month = "4",
day = "1",
doi = "10.1016/j.disc.2018.12.009",
language = "English",
volume = "342",
pages = "1098--1107",
journal = "Discrete Mathematics",
issn = "0012-365X",
publisher = "Elsevier",
number = "4",

}

TY - JOUR

T1 - Inherited conics in Hall planes

AU - Blokhuis, Aart

AU - Kovács, István

AU - Nagy, Gábor P.

AU - Szőnyi, T.

PY - 2019/4/1

Y1 - 2019/4/1

N2 - The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

AB - The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of PG(2,q) remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.

KW - Arcs

KW - Finite projective planes

KW - Hall planes

UR - http://www.scopus.com/inward/record.url?scp=85059816790&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85059816790&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2018.12.009

DO - 10.1016/j.disc.2018.12.009

M3 - Article

VL - 342

SP - 1098

EP - 1107

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 4

ER -