Monatshefte für Mathematik

, Volume 93, Issue 3, pp 211–223

On cubic polynomials III. Systems ofp-adic equations

  • Wolfgang M. Schmidt


We study systems of equations
$$\mathfrak{F}_1 = \ldots = \mathfrak{F}_r = 0,$$
where\(\mathfrak{F}_1 , \ldots ,\mathfrak{F}_r \) are cubic forms withp-adic coefficients. Such a system has a nontrivialp-adic solution if the number of variables is at least 50,000 ·r3. Further we will give estimates for the number of solutions of certain systems of cubic congruences.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Birch, B. J., Lewis, D. J., Murphy, T. G.: Simultaneous quadratic forms. Amer. J. Math.84, 110–115 (1962).Google Scholar
  2. [2]
    Brauer, R.: A note on systems of homogeneous algebraic equations. Bull. Amer. Math. Soc.51, 749–755 (1945).Google Scholar
  3. [3]
    Dem'janov, V. B.: On cubic forms in discretely normed fields. (Russian.) Dokl. Akad. Nauk SSSR (NS)74, 889–891 (1950).Google Scholar
  4. [4]
    Dem'janov, V. B.: Pairs of quadratic forms over a complete field with discrete norm with a finite field of residue classes (Russian.) Izv. Akad. SSSR ser. Mat.20, 307–324 (1956).Google Scholar
  5. [5]
    Leep, D.: Systems of quadratic forms. (In preparation.)Google Scholar
  6. [6]
    Leep, D., Schmidt, W. M.: Systems of homogeneous equations. (In preparation.)Google Scholar
  7. [7]
    Lewis, D. J.: Cubic homogeneous polynomials overp-adic number fields. Ann. Math. (2)56, 473–478 (1952).Google Scholar
  8. [8]
    Lewis, D. J.: Diophantine equations:p-adic methods. Studies in Number Theory, pp. 25–75. Math. Assoc. Amer. Englewood Cliffs, N. J.: Prentice Hall. 1969.Google Scholar
  9. [9]
    Schmidt, W. M.: Equations over Finite Fields. Lecture Notes Math. 536. Berlin-Heidelberg-New York: Springer. 1976.Google Scholar
  10. [10]
    Schmidt, W. M.: Simultaneousp-adic zeros of quadratic forms. Mh. Math.90, 45–65 (1980).Google Scholar
  11. [11]
    Schmidt, W. M.: On cubic polynomials. II. Multiple exponential sums. Mh. Math.93, 141–168 (1982).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Wolfgang M. Schmidt
    • 1
  1. 1.Mathematics DepartmentUniversity of ColoradoBoulderU.S.A.

Personalised recommendations