Hyperbolic ovals in finite planes
type: Item
instance of: scholarly article
title: Hyperbolic ovals in finite planes
zbMATH Open document ID: 1061.51008
author: Yanyan Li
published in: Designs, Codes and Cryptography
publication date: 2004-08-16T00:00:00Z
review text: Let \(n = p^{2h}\) be a square power of an odd prime \(p\), and \(GF(n)\) the finite field of order \(n\). The triple \((N(n), +, \odot)\), where \(N(n)\) is the set of the elements of \(GF(n),\) and \(+, \odot\) are two operations on \(N(n)\) such that \(+\) is the same as the addition in \(GF(n)\), and the multiplication \(\odot\) is defined by \[ x \odot y = \begin{cases} xy & \text{for square } x\in GF(n), \\ x y^{p^h} & \text{for non-square } x\in GF(n), \end{cases} \] is a nearfield of dimension \(2\) over its center. Denote such a nearfield with \(N(n)\). Let \(\alpha\) be the affine plane of order \(n\) coordinatized by the above nearfield \(N(n)\). For any \(a, b, c, d\in GF(n)\) such that \(c\neq 0\) and \(a d - b c \neq 0\) let \[ \Omega_{a,b,c,d} = \biggl\{P(x, y)\in \alpha \, \left| \, x\in GF(n),\;x\neq -d/c,\;y = \frac{a x + b}{c x + d}\right.\biggr\}. \] Clearly \(\Omega_{a,b, c, d}\) has size \(n - 1\). \textit{L. A. Rosati} [Boll. Unione Mat. Ital. (4) 4, 463--467 (1971; Zbl 0229.50027)] proved the following result. Theorem. \(\Omega_{a,b,c,d}\) is a hyperbolic oval whose infinite points are \(X_\infty\) and \(Y_\infty\). Such hyperbolic ovals are equivalent under the collineation group of \(\pi\). In the paper under review the authors use the term ``Rosati oval'' to indicate \(\Omega_{a,b,c,d}\) and they give a characterization of such an oval. They prove the following result. Theorem. Let \(\Omega\) be a transitive hyperbolic oval in an affine translation plane \(\pi\) of odd order \(n\). Let \(D\) be a dihedral collineation group of order \(2(n - 1)\) of \(\pi\). If \(D\) preserves \(\Omega\) and acts transitively on the affine points of \(\Omega\), then one of the following cases occurs. i. \(\pi\) is the Desarguesian plane and \(\Omega\) is a conic. All non-central involutions are affine homologies, or ii. \(\pi\) is the regular nearfield plane of dimension \(2\) over its center and \(\Omega\) is a Rosati oval. The group \(D\) has \(\frac{1}{2…
reviewed by: V. Napolitano
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 2091520
zbMATH Keywords: Oval
zbMATH Keywords: projective plane
zbMATH Keywords: nearfield
Wikidata QID: Q59249236
MaRDI profile type: Publication
DOI: 10.1023/B:DESI.0000029227.72102.ED
description: scientific article; zbMATH DE number 2091520