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Sziklai’s conjecture on the number of points of a plane curve over a finite field. II. (English) Zbl 1211.14037
McGuire, Gary (ed.) et al., Finite fields. Theory and applications. Proceedings of the 9th international conference on finite fields and applications, Dublin, Ireland, July 13–17, 2009. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4786-2/pbk). Contemporary Mathematics 518, 225-234 (2010).
Let $$C$$ be a projective plane curve of degree $$d$$ over $$\mathbb{F}_q$$ with no $$\mathbb{F}_q$$-linear components, and denote the set of such curves by $${\mathcal{C}}(q,d)$$. Sziklai posed in his paper [P. Sziklai, Finite Fields Appl. 14, No. 1, 41–43 (2008; Zbl 1185.14017)] a conjecture to the effect that (*) for any $$C\in {\mathcal{C}}(q,d)$$, the number of $$\mathbb{F}_q$$-rational points $$N_q(C)$$ of $$C$$ would be at most $$(d-1)q+1$$. The authors of this paper notice, however, that for the validity of the conjecture one must exclude an explicit example of a curve $$C_0$$ of degree four over $$\mathbb{F}_4$$ with 14 $$\mathbb{F}_4$$-rational points. They call the assertion (*) with $${\mathcal{C}}(q,d)$$ replaced by $${\mathcal{C}}(q,d)-($$curves projectively equivalent to $$C_0)$$ the modified Sziklai’s conjecture. Furthermore they remark that the conjecture makes sense only if $$2\leq d\leq q+1$$ because the conjectural bound exceeds the obvious upper bound $$N_q(\mathbb{P}^2(\mathbb{F}_q))=q^2+q+1$$ when $$d\geq q+2$$. Thereafter they prove the main result of the paper which shows that if $$d=q$$, then the modified Sziklai’s conjecture holds true.
[For part I, cf. Finite Fields Appl. 15, No. 4, 468–474 (2009; Zbl 1194.14031).]
For the entire collection see [Zbl 1193.11003].

MSC:
 14H50 Plane and space curves 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
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