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The uniqueness of a plane curve of degree \(q\) attaining Sziklai’s bound over \(\mathbb F_{q}\). (English) Zbl 1243.14024
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 \(N_q(C)\) 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 14 \(F_4\)-rational points. This conjecture 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 are interested in those curves of degree \(d=q\) attaining equality in \((*)\). In fact this problem has already been considered by the authors for \(d=q+1\) [Finite Fields Appl. 17, No. 3, 240–253 (2011; Zbl 1215.14033)] and \(d\geq q+2\) [“Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: Supplements to a work of Tallini”, arXiv:0903.1918]. There is just one curve, up to projective equivalence, of degree \(q\) namely \(X^{q}-XZ^{q-1}+X^{q-1}Y-Y^q=0\) whose number of \({\mathbb F}_q\)-rational points equals \((q-1)q+1\).

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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