Toward determination of optimal plane curves with a fixed degree over a finite field
type: Item
instance of: scholarly article
title: Toward determination of optimal plane curves with a fixed degree over a finite field
zbMATH Open document ID: 1215.14033
author: Masaaki Homma
author: Seon Jeong Kim
published in: Finite Fields and their Applications
publication date: 2011-05-31T00:00:00Z
review text: Let \(C\) be a projective plane curve of degree \(d\) defined over the finite field \(k\) with \(q\) elements. The curve \(C\) may be reducible but with no \(k\)-linear components. Let \(N_q(C)\) be the number of \(k\)-rational points of \(C\). \textit{P. Sziklai} [Finite Fields Appl. 14, No. 1, 41--43 (2008; Zbl 1185.14017)] conjectured that \(N_q(C)\leq (d-1)q+1\), except for the curve \[ X^4+Y^4+Z^4+X^2Y^2+Y^2Z^2+Z^2X^2+X^2YZ+XY^2Z+XYZ^2=0 \] over \({\mathbb F}_4\). This result was proved by \textit{M. Homma} and \textit{S. J. Kim} [Finite Fields Appl. 16, No. 5, 315--319 (2010; Zbl 1196.14030)]. In the paper under review, the authors deal with the case \(d=q+1\) and determine the curves whose number of \(k\)-rational points equals \(q^2+1\). For \(q\geq 5\) or \(q=2\) there is just one curve, up to projective equivalence, namely \(X^{q+1}-X^2Z^{q-1}+Y^qZ-YZ^q=0\). For \(q=3\) and for \(q=4\) there are two such curves, up to projective equivalence.
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14N10
zbMATH DE Number: 5902690
zbMATH Keywords: plane curve
zbMATH Keywords: finite field
zbMATH Keywords: rational point
describes a project that uses: manYPoints
MaRDI profile type: Publication
full work available at URL: j.ffa.2010.12.003
OpenAlex ID: W2094305993
cites work: Quadratic forms, generalized Hamming weights of codes and curves with many points
cites work: Correction to ``Frobenius non-classical curves''
cites work: Q4370894
cites work: Q5386122
cites work: Around Sziklai's conjecture on the number of points of a plane curve over a finite field
cites work: Sziklai's conjecture on the number of points of a plane curve over a finite field II
cites work: Sziklai's conjecture on the number of points of a plane curve over a finite field. III
cites work: Weierstrass Points and Curves Over Finite Fields
cites work: A bound on the number of points of a plane curve
cites work: Q3292942
DOI: 10.1016/J.FFA.2010.12.003
description: scientific article; zbMATH DE number 5902690