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Other literature type . Preprint . 2020

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https://doi.org/10.48550/arxiv...

Article . 2020

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A bound for the number of points of space curves over finite fields

Authors: Beelen, Peter; Montanucci, Maria;

A bound for the number of points of space curves over finite fields

Abstract

For a non-degenerate irreducible curve $C$ of degree $d$ in $\mathbb{P}^3$ over $\mathbb{F}_q$, we prove that the number $N_q(C)$ of $\mathbb{F}_q$-rational points of $C$ satisfies the inequality $N_q(C) \leq (d-2)q+1$. Our result improves the previous bound $N_q(C) \leq (d-1)q+1$ obtained by Homma in 2012 and leads to a natural conjecture generalizing Sziklai's bound for the number of points of plane curves over finite fields.

Keywords

Mathematics - Algebraic Geometry, FOS: Mathematics, Algebraic Geometry (math.AG)

Mathematics - Algebraic Geometry, FOS: Mathematics, Algebraic Geometry (math.AG)

i(π ∩ C, P ) ≥ 4.

[1] E. Ballico, Special inner projections of projective varieties, Ann. Univ. Ferrara 50, 23-26 (2004). [OpenAIRE]

[2] V. Bayer and A. Hefez, Strange curves, Comm. Algebra 19, 3041-3059 (1991). [OpenAIRE]

[3] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52. Springer-Verlag, New YorkHeidelberg, (1977).

[4] J.W.P. Hirschfeld, G. Korchm´aros and F. Torres, Algebraic Curves over a Finite Field, Princeton Series in Applied Mathematics, Princeton, (2008).

[5] M. Homma, A bound on the number of points of a curve in projective space over a finite field, Theory and Applications of Finite Fields in: Contemp. Mat. 579, 103-110 (2012).

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

[7] M. Homma and S.J. Kim, Sziklai's conjecture on the number of points of a plane curve over a finite field II, in: G. McGuire, G.L. Mullen, D. Panario, I.E. Shparlinski (Eds.), Finite Fields: Theory and Applications, in: Contemp. Math. 518, AMS, Providence, 225-234 (2010).

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

[9] P. Samuel, Lectures on old and new results on algebraic curves (notes by Anantharaman), Tata Inst. Fund. Res., (1966).

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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).

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!**

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