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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.
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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