Found an issue? Give us feedback

arXiv.org e-Print Archive

Other literature type . Preprint . 2022

Data sources: arXiv.org e-Print Archive

Finite Fields and Their Applications

Article . 2023 . Peer-reviewed

License: Elsevier TDM

Data sources: Crossref

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__

Optimal plane curves of degree q − 1 over a finite field

Authors: Walteir de Paula Ferreira; Pietro Speziali;

Optimal plane curves of degree q − 1 over a finite field

Abstract

Let $q\geq 5$ be a prime power. In this note, we prove that if a plane curve $\mathcal{X}$ of degree $q - 1$ defined over $\mathbb{F}_q$ without $\mathbb{F}_q$-linear components attains the Sziklai upper bound $(d-1)q+1 = (q - 1)^2$ for the number of its $\mathbb{F}_q$-rational points, then $\mathcal{X}$ is projectively equivalent over $\mathbb{F}_q$ to the curve $ \mathcal{C}_{(\alpha,\beta,\gamma)} : \alpha X^{q - 1} + \beta Y^{q - 1} + \gamma Z^{q - 1} = 0$ for some $\alpha, \beta, \gamma \in \mathbb{F}_q^{*}$ such that $\alpha + \beta + \gamma = 0$. This completes the classification of curves that are extremal with respect to the Sziklai bound. Also, since the Sziklai bound is equal to the St\"ohr-Voloch's bound for plane curves of degree $q - 1$, our main result classifies the $\mathbb{F}_q$-Frobenius classical extremal plane curves of degree $q - 1$.

Keywords

Algebra and Number Theory, Applied Mathematics, General Engineering, Theoretical Computer Science, Mathematics - Algebraic Geometry, FOS: Mathematics, Algebraic Geometry (math.AG), 114H37, 14H05

Algebra and Number Theory, Applied Mathematics, General Engineering, Theoretical Computer Science, Mathematics - Algebraic Geometry, FOS: Mathematics, Algebraic Geometry (math.AG), 114H37, 14H05

[1] N. Arakelian, S. Tafazolian, F. Torres, On the spectrum for the genera of maximal curves over small fields. Advances in Mathematics of Communications, 12 (2018), 143-149.

[2] J. Bierbrauer, Introduction to coding theory (2nd ed.). Chapman and Hall/CRC, 2016.

[3] I. Bouyukliev, E.J. Cheon, T. Maruta, T. Okazaki, On the (29, 5)-Arcs in PG(2, 7) and Some Generalized Arcs in PG(2, q), Mathematics, 8 (2020), 320.

[4] M. Datta, Maximum number of Fq-rational points on nonsingular threefolds in P4. Finite Fields and Their Applications, 59 (2019), 86-96.

[5] A. Hefez, J.F. Voloch, Frobenius nonclassical curves. Arch. Math, 54 (1990), 263-273.

[6] J.W.P Hirschfeld, G. Korchma´ros and F. Torres, Algebraic Curves over a Finite Field, Princeton Univ. Press, Princeton, (2008), xiii + 720 pp.

[7] J.W.P. Hirschfeld, L. Storme, J.A. Thas, A characterization of Hermitian curves, J Geom 41 (1991), 72-78.

[8] J.W.P. Hirschfeld, Projective geometries over finite fields, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1979.

[9] 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]

[10] 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. Linear Algebra and its Applications, 438 (2013), 69-985.

- 2014IsAmongTopNSimilarDocuments
- 2021IsAmongTopNSimilarDocuments
- 2012IsAmongTopNSimilarDocuments
- 2020IsAmongTopNSimilarDocuments
- 2022IsAmongTopNSimilarDocuments
- 2013IsAmongTopNSimilarDocuments
- 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).0 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).0 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!**

0

Average

Average

Average

Green

Fields of Science (6) View all & suggest

Fields of Science