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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
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References:
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