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Toward determination of optimal plane curves with a fixed degree over a finite field. (English) Zbl 1215.14033
Let $$C$$ be a projective plane curve of degree $$d$$ defined over the finite field $$k$$ with $$q$$ elements. The curve $$C$$ may be reducible but with no $$k$$-linear components. Let $$N_q(C)$$ be 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 $X^4+Y^4+Z^4+X^2Y^2+Y^2Z^2+Z^2X^2+X^2YZ+XY^2Z+XYZ^2=0$ over $${\mathbb F}_4$$. This result was proved by M. Homma and S. J. Kim [Finite Fields Appl. 16, No. 5, 315–319 (2010; Zbl 1196.14030)].
In the paper under review, the authors deal with the case $$d=q+1$$ and determine the curves whose number of $$k$$-rational points equals $$q^2+1$$. For $$q\geq 5$$ or $$q=2$$ there is just one curve, up to projective equivalence, namely $$X^{q+1}-X^2Z^{q-1}+Y^qZ-YZ^q=0$$. For $$q=3$$ and for $$q=4$$ there are two such curves, up to projective equivalence.

##### 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:
plane curve; finite field; rational point
manYPoints
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##### References:
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