Monatshefte für Mathematik

, Volume 93, Issue 4, pp 329–348

On cubic polynomials

IV. systems of rational equations
  • Wolfgang M. Schmidt


It is shown that a system ofr homogeneous cubic equations with rational coefficients has a nontrivial solution in rational integers if the number of variables is at least (10r)5. For most such systems, an asymptotic formula holds for the numberzPof solutions whose components have modulus <P.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Birch, B. J.: Homogeneous forms of odd degree in a large number of variables. Mathematika4, 102–105 (1957).Google Scholar
  2. [2]
    Birch, B. J.: Forms in many variables. Proc. Royal. Soc. A265, 245–263 (1962).Google Scholar
  3. [3]
    Borsuk, K.: Drei Sätze über dien-dimensionale Euklidische Sphäre. Fund. Math.20, 177–190 (1933).Google Scholar
  4. [4]
    Brauer, R.: A note on systems of homogeneous algebraic equations. Bull. Amer. Math. Soc.51, 749–755 (1945).Google Scholar
  5. [5]
    Davenport, H.: Cubic forms in 32 variables. Phil Trans. Royal Soc. London A251, 193–232 (1959).Google Scholar
  6. [6]
    Davenport, H.: Cubic forms in 16 variables. Proc. Royal Soc. A272, 285–303 (1963).Google Scholar
  7. [7]
    Davenport, H., Lewis, D. J.: Exponential sums in many variables. Amer. J. Math.84, 649–665 (1962).Google Scholar
  8. [8]
    Davenport, H., Lewis, D. J.: Non-homogeneous cubic equations. J. London Math Soc.39, 657–671 (1964).Google Scholar
  9. [9]
    Davenport, H., Lewis, D. J.: Simultaneous equations of additive type. Phil. Trans. Royal Soc. London A264, 557–595 (1969).Google Scholar
  10. [10]
    Dam'janov, V. B.: On cubic forms in discretely normed fields. Akad. Nauk SSSR. (NS)74, 889–891 (1950).Google Scholar
  11. [11]
    Leep, D., Schmidt, W. M.: Systems of homogeneous equations. (In preparation.)Google Scholar
  12. [12]
    Lewis, D. J.: Cubic homogeneous polynomials overp-adic number fields. Ann. Math. (2)56, 473–478 (1952).Google Scholar
  13. [13]
    Schmidt, W. M.: Simultaneousp-adic zeros of quadratic forms. Mh. Math.90, 45–65 (1980).Google Scholar
  14. [14]
    Schmidt, W. M.: Simultaneous rational zeros of quadratic forms. Progress Math. (In print.)Google Scholar
  15. [15]
    Schmidt, W. M.: On cubic polynomials III. Systems ofp-adic equations. Mh. Math.93, 211–223 (1982).Google Scholar
  16. [16]
    Tartakovskii, V. A.: The number of representations of large numbers by a form of “general type” with many variables. (Russian). I. Vestnik Leningrad Univ.13, no. 7, 131–154 (1958), and II. ibid.14, no. 7, 5–17 (1959).Google Scholar
  17. [17]
    Tartakovskii, V. A.: On representability of large numbers by forms of “general type” with a large number of variables. (Russian). Izv. Vysš. Učebn. Zaved. Matematika1959, no. 1 (2), 161–173 (1959).Google Scholar
  18. [18]
    Watson, G. L.: Non-homogeneous cubic equations. Proc. London Math. Soc. (3)17, 271–295 (1967).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Wolfgang M. Schmidt
    • 1
  1. 1.Mathematics DepartmentUniversity of ColoradoBoulderUSA

Personalised recommendations