Found an issue? Give us feedback

Finite Fields and Their Applications

Article . 2012 . Peer-reviewed

License: Elsevier Non-Commercial

Please grant OpenAIRE to access and update your ORCID works.

This Research product is the result of merged Research products in OpenAIRE.

You have already added 0 works in your ORCID record related to the merged Research product.

You have already added 0 works in your ORCID record related to the merged Research product.

This Research product is the result of merged Research products in OpenAIRE.

You have already added 0 works in your ORCID record related to the merged Research product.

You have already added 0 works in your ORCID record related to the merged Research product.

All Research products

```
<script type="text/javascript">
<!--
document.write('<div id="oa_widget"></div>');
document.write('<script type="text/javascript" src="https://www.openaire.eu/index.php?option=com_openaire&view=widget&format=raw&projectId=undefined&type=result"></script>');
-->
</script>
```

For further information contact us at __helpdesk@openaire.eu__

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

- Kanazawa University Japan
- Kanagawa University Japan
- Gyeongsang National University Korea (Republic of)

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

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

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

[1] Y. Aubry, M. Perret, A Weil theorem for singular curves, in: R. Pellikaan, M. Perret, S. Vla˘dut¸ (Eds.), Arithmetic Geometry and Coding Theory, Luminy, 1993, de Gruyter, Berlin, 1996, pp. 1-7. [OpenAIRE]

[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]

- 2023IsAmongTopNSimilarDocuments
- 2012IsAmongTopNSimilarDocuments
- 2013IsAmongTopNSimilarDocuments
- 2014IsAmongTopNSimilarDocuments
- 2021IsAmongTopNSimilarDocuments
- 2020IsAmongTopNSimilarDocuments
- 2022IsAmongTopNSimilarDocuments
- 2020IsAmongTopNSimilarDocuments
- 2016IsAmongTopNSimilarDocuments

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).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 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).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 Powered byBIP!

Found an issue? Give us feedback

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).

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.

Popularity provided by **BIP!**

3

Average

Average

Average

hybrid

Fields of Science (7) View all & suggest

Fields of Science