An elementary proof of a partial improvement to the Ax-Katz Theorem

  • O. Moreno
  • C. J. Moreno
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 673)


Moreno and Moreno have recently given a result that in many cases improves upon the Ax-Katz theorem. We will presently give an elementary and self contained proof of this and also give a corresponding improvement upon a result of Adolphson and Sperber.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Adolphson and S. Sperber, p-adic estimates for exponential sums and the theorem of Chevalley-Warning, Ann. Scient. E. N. Superior 4th. series 20 (1987), 545–556.Google Scholar
  2. [2]
    E. Artin, Collected Papers, Springer-Verlag, New York, 1965.Google Scholar
  3. [3]
    J. Ax, Zeros of polynomials over finite fields, Amer. J. Math. 86 (1964), 255–261.Google Scholar
  4. [4]
    C. Chevalley, Demonstration d'une hypothese de M. Artin, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11 (1936), 73–75.Google Scholar
  5. [5]
    B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math. 82 (1960), 631–648.Google Scholar
  6. [6]
    N. M. Katz, On a theorem of Ax, Amer. J. Math. 93 (1971), 485–499.Google Scholar
  7. [7]
    P. V. Kumar and O. Moreno, Prime-Phase Sequences with Periodic Correlation Properties Better Than Binary Sequences, IEEE Trans. Inform. Theory Vol. 37 No. 3 (1991), 603–616.Google Scholar
  8. [8]
    R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Math and its Appl. 20 Addison-Wesley Publishing Company, 1983.Google Scholar
  9. [9]
    S. Litsyn, C. J. Moreno and O. Moreno, Divisibility properties and new bounds for exponential sums in one and several variables, submitted to the AAECC Journal.Google Scholar
  10. [10]
    F. J. MacWilliams and N. J. A. Sloane, The theory of error correcting codes, North-Holland, Amsterdam, 1977.Google Scholar
  11. [11]
    Oscar Moreno and Carlos Moreno, Improvement of the Chevalley Warning and the Ax-Katz Theorems, accepted in the American Journal of Math. Google Scholar
  12. [12]
    O. Moreno and C. J. Moreno, The MacWilliams-Sloane Conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes, submitted to the IEEE IT Trans. Google Scholar
  13. [13]
    Daqing Wan, An elementary proof of a theorem of Katz, American Journal of Mathematics 111 (1989), 1–8.Google Scholar
  14. [14]
    E. Warning, Bermerkung zur vorstehenden Arbeit von Herrn Chevalley, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 11 (1936), 76–83.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • O. Moreno
    • 1
  • C. J. Moreno
    • 2
  1. 1.University of Puerto Rico Rio PiedrasUSA
  2. 2.BaruchCUNYN. SalemUSA

Personalised recommendations