The uniqueness of a plane curve of degree \(q\) attaining Sziklai’s bound over \(\mathbb F_{q}\).

*(English)*Zbl 1243.14024Let \(C\) 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 \((q-1)q+1\).

Reviewer: Fernando Torres (Campinas)

##### MSC:

14H50 | Plane and space curves |

14G15 | Finite ground fields in algebraic geometry |

14G05 | Rational points |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

PDF
BibTeX
XML
Cite

\textit{M. Homma} and \textit{S. J. Kim}, Finite Fields Appl. 18, No. 3, 567--580 (2012; Zbl 1243.14024)

Full Text:
DOI

##### References:

[1] | Aubry, Y.; Perret, M., A Weil theorem for singular curves, (), 1-7 · Zbl 0873.11037 |

[2] | Hirschfeld, J.W.P., Projective geometries over finite fields, (1998), Oxford University Press Oxford · Zbl 0899.51002 |

[3] | Homma, M.; Kim, S.J., Nonsingular plane filling curves of minimum degree over a finite field and their automorphism groups: supplements to a work of tallini, (2009) |

[4] | Homma, M.; Kim, S.J., Around sziklaiʼs conjecture on the number of points of a plane curve over a finite field, Finite fields appl., 15, 468-474, (2009) · Zbl 1194.14031 |

[5] | Homma, M.; Kim, S.J., Sziklaiʼs conjecture on the number of points of a plane curve over a finite field II, Contemp. math., 518, 225-234, (2010) · Zbl 1211.14037 |

[6] | Homma, M.; Kim, S.J., Sziklaiʼs conjecture on the number of points of a plane curve over a finite field III, Finite fields appl., 16, 315-319, (2010) · Zbl 1196.14030 |

[7] | Homma, M.; Kim, S.J., Toward determination of optimal plane curves with a fixed degree over a finite field, Finite fields appl., 17, 240-253, (2011) · Zbl 1215.14033 |

[8] | Schoof, R., Nonsingular plane cubic curves over finite fields, J. combin. theory ser. A, 46, 183-211, (1987) · Zbl 0632.14021 |

[9] | Sziklai, P., A bound on the number of points of a plane curve, Finite fields appl., 14, 41-43, (2008) · Zbl 1185.14017 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.