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On a number of rational points on a plane curve of low degree. (English) Zbl 1386.14120
Let $$C$$ be a projective plane curve of degree $$d\geq 2$$ defined over the finite field $$k$$ with $$q$$ elements. The curve $$C$$ may be reducible but with no $$k$$-linear components. Let $$N_q(C)$$ denote the number of $$k$$-rational points of $$C$$. P. Sziklai [Finite Fields Appl. 14, No. 1, 41–43 (2008; Zbl 1185.14017)] conjectured that $$N_q(C)\leq (d-1)q+1\, (*)$$ except for the curve over $${\mathbb F}_4$$ $$X^4+Y^4+Z^4+X^2Y^2+Y^2Z^2+Z^2X^2+X^2YZ+XY^2Z+XYZ^2=0$$ which has $$(d-1)q+2=14$$ $${\mathbb F}_4$$-rational points. This conjecture was proved by M. Homma and S. Kim [Finite Fields Appl. 16, No. 5, 315–319 (2010; Zbl 1196.14030)]. The case where equality holds in $$(*)$$ with $$d\in\{q+2,q+1,q,q-1,\sqrt{q}-1,2\}$$ where studied in detail by the same authors e.g. in [Finite Fields Appl. 17, No. 3, 240–253 (2011; Zbl 1215.14033); ibid. 18, No. 3, 567–586 (2012; Zbl 1243.14024)]. In general there is just one curve attaining such a bound in this case.
In the paper under review, the authors investigate Skilais’s bound $$(*)$$ above for $$q\leq 7$$ and $$d$$ small. For example for a curve $$C$$ of degree $$5$$, $$N_7(C)\leq 26<(d-1)q+1$$ and the bound is attained by several curves.
##### MSC:
 14H50 Plane and space curves 14G15 Finite ground fields in algebraic geometry 14G05 Rational points 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
##### Keywords:
finite field; plane curve; rational point
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##### References:
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