Around Sziklai's conjecture on the number of points of a plane curve over a finite field
type: Item
instance of: scholarly article
title: Around Sziklai's conjecture on the number of points of a plane curve over a finite field
zbMATH Open document ID: 1194.14031
author: Masaaki Homma
author: Seon Jeong Kim
published in: Finite Fields and their Applications
publication date: 2009-09-15T00:00:00Z
review text: The paper considers the problem of bounding \(M_q(d)\), the maximal number of rational points on a plane projective curve of degree \(d\) over the finite field with \(q\) elements that does not contain a line. It shows that the question is interesting only when \(2\leq d\leq q+1\). A conjecture by Sziklai states that \(M_q(d)\leq (d-1)q+1\), and it is known that \(M_q(d)\leq (d-1)q+\lfloor d/2\rfloor\). The authors prove that \(M_q(d)\leq (d-1)q+(q+2-d)\), which is for many \((d,q)\) sharper than the best bound known before, and implies Sziklai's conjecture for \(d=q+1\). The proof uses a very nice `two ways of counting' the elements of a certain set. The paper also disproves the general conjecture by showing that \(M_4(4)=14\), a bound which is attained by essentially just one curve.
reviewed by: Hendrik Hubrechts
Mathematics Subject Classification ID: 14G15
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 5603986
zbMATH Keywords: many points
zbMATH Keywords: plane curves
zbMATH Keywords: bound
describes a project that uses: manYPoints
MaRDI profile type: Publication
full work available at URL: j.ffa.2009.02.008
OpenAlex ID: W1966931331
Wikidata QID: Q123352873
cites work: Q4004114
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cites work: Q3292942
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DOI: 10.1016/J.FFA.2009.02.008
description: scientific article; zbMATH DE number 5603986