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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.

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