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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

Prime Ideal Local Ring Prime Divisor Plane Curve Irreducible Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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