Toward determination of optimal plane curves with a fixed degree over a finite field
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 \(C\). \textit{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 \textit{M. Homma} and \textit{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.
scientific article; zbMATH DE number 5902690