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On a number of rational points on a plane curve of low degree. (English) Zbl 1386.14120
Let \(C\) be a projective plane curve of degree \(d\geq 2\) 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 \((d-1)q+2=14\) \({\mathbb F}_4\)-rational points. This conjecture was proved by M. Homma and S. Kim [Finite Fields Appl. 16, No. 5, 315–319 (2010; Zbl 1196.14030)]. The case where equality holds in \((*)\) with \(d\in\{q+2,q+1,q,q-1,\sqrt{q}-1,2\}\) where studied in detail by the same authors e.g. in [Finite Fields Appl. 17, No. 3, 240–253 (2011; Zbl 1215.14033); ibid. 18, No. 3, 567–586 (2012; Zbl 1243.14024)]. In general there is just one curve attaining such a bound in this case.
In the paper under review, the authors investigate Skilais’s bound \((*)\) above for \(q\leq 7\) and \(d\) small. For example for a curve \(C\) of degree \(5\), \(N_7(C)\leq 26<(d-1)q+1\) and the bound is attained by several curves.
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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