Plane Curves

  • Michael D. Fried
  • Moshe Jarden
Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE3, volume 11)

Abstract

Here (Corollary 4.8) we use Theorem 3.14 to prove the following: If f (X,Y) is an absolutely irreducible polynomial of degree d defined over a field K of q elements,then the number N of K-rational solutions to the equation f (X, Y)= 0 \(\left| {N - (q - 1)} \right|(d - 1)(d - 2)\sqrt q + d.\)

Keywords

Hull 

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Notes

  1. [S7]
    J-P Serre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini,C.R. Acad. Sci. Paris sèrie I, 296 (1983), 397–402Google Scholar
  2. [DV]
    V.G. Drinfeld and S. Vladut, Number of points on an algebraic curve,Functional analysis and its applications 17 (1983), 68–69 (russian)Google Scholar
  3. [TVZ]
    M.A. Tsfasman, S.G. Vladut and T. Zink, Modular curves, Shimura curves, and Goppa codes better that Warshamov-Gilbert bound, Mathematische Nachrichten 109 (1982), 21–28MathSciNetMATHCrossRefGoogle Scholar
  4. [Gp]
    V.D. Goppa, Codes on algebraic curves, Soviet Mathematics Doklady 24 (1981), 170–172Google Scholar
  5. [LN]
    R. Lidl and H. Niederriter, Finite fields, Encyclopedia of Mathematics and its applications 20, Addison-Wesley, Reading, 1983Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Michael D. Fried
    • 1
  • Moshe Jarden
    • 2
  1. 1.Mathematical DepartmentUniversity of FloridaGainsvilleUSA
  2. 2.School of Mathematical Sciences, Raymond and Beverly Sackler, Faculty of Exact SciencesTel Aviv UniversityRamat Aviv, Tel AvivIsrael

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