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Finite Fields and Their Applications
Article . 2012 . Peer-reviewed
License: Elsevier Non-Commercial
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The uniqueness of a plane curve of degree q attaining Sziklaiʼs bound over Fq

Authors: Homma, Masaaki; Kim, Seon Jeong;

The uniqueness of a plane curve of degree q attaining Sziklaiʼs bound over Fq

Abstract

Abstract We prove the uniqueness of a plane curve of degree q over a finite field F q which attains Sziklaiʼs bound q ( q − 1 ) + 1 . More precisely, if a plane curve of degree q over F q has q ( q − 1 ) + 1 rational points, then it is projectively equivalent to the curve defined by the equation X q − X Z q − 1 + X q − 1 Y − Y q = 0 . Although the case q = 4 is the exception to Sziklaiʼs bound, the uniqueness of a curve of degree 4 with 13 points over F 4 still holds.

Related Organizations
Subjects by Vocabulary

Microsoft Academic Graph classification: Circular algebraic curve Quartic plane curve Plane curve Butterfly curve (algebraic) Mathematical analysis Osculating curve Combinatorics Finite field Rational point Uniqueness Mathematics

Keywords

Algebra and Number Theory, Applied Mathematics, General Engineering, Finite field, Theoretical Computer Science, Plane curve, Rational point

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[2] J.W.P. Hirschfeld, Projective Geometries Over Finite Fields, second edition, Oxford University Press, Oxford, 1998.

[3] M. Homma, S.J. Kim, Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini, arXiv:0903.1918, 2009.

[4] M. Homma, S.J. Kim, Around Sziklai's conjecture on the number of points of a plane curve over a finite field, Finite Fields Appl. 15 (2009) 468-474. [OpenAIRE]

[5] M. Homma, S.J. Kim, Sziklai's conjecture on the number of points of a plane curve over a finite field II, Contemp. Math. 518 (2010) 225-234.

[6] M. Homma, S.J. Kim, Sziklai's conjecture on the number of points of a plane curve over a finite field III, Finite Fields Appl. 16 (2010) 315-319. [OpenAIRE]

[7] M. Homma, S.J. Kim, Toward determination of optimal plane curves with a fixed degree over a finite field, Finite Fields Appl. 17 (2011) 240-253.

[8] R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987) 183-211. [OpenAIRE]

[9] P. Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008) 41-43. [OpenAIRE]

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    3
    popularity
    This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
    Average
    influence
    This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
    Average
    impulse
    This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
    Average
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citations
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
3
Average
Average
Average
hybrid