Skip to main content
SpringerLink
Log in

On cubic polynomials

IV. systems of rational equations

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

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 numberz P of solutions whose components have modulus <P.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birch, B. J.: Homogeneous forms of odd degree in a large number of variables. Mathematika4, 102–105 (1957).

    Google Scholar 

  2. Birch, B. J.: Forms in many variables. Proc. Royal. Soc. A265, 245–263 (1962).

    Google Scholar 

  3. Borsuk, K.: Drei Sätze über dien-dimensionale Euklidische Sphäre. Fund. Math.20, 177–190 (1933).

    Google Scholar 

  4. Brauer, R.: A note on systems of homogeneous algebraic equations. Bull. Amer. Math. Soc.51, 749–755 (1945).

    Google Scholar 

  5. Davenport, H.: Cubic forms in 32 variables. Phil Trans. Royal Soc. London A251, 193–232 (1959).

    Google Scholar 

  6. Davenport, H.: Cubic forms in 16 variables. Proc. Royal Soc. A272, 285–303 (1963).

    Google Scholar 

  7. Davenport, H., Lewis, D. J.: Exponential sums in many variables. Amer. J. Math.84, 649–665 (1962).

    Google Scholar 

  8. Davenport, H., Lewis, D. J.: Non-homogeneous cubic equations. J. London Math Soc.39, 657–671 (1964).

    Google Scholar 

  9. Davenport, H., Lewis, D. J.: Simultaneous equations of additive type. Phil. Trans. Royal Soc. London A264, 557–595 (1969).

    Google Scholar 

  10. Dam'janov, V. B.: On cubic forms in discretely normed fields. Akad. Nauk SSSR. (NS)74, 889–891 (1950).

    Google Scholar 

  11. Leep, D., Schmidt, W. M.: Systems of homogeneous equations. (In preparation.)

  12. Lewis, D. J.: Cubic homogeneous polynomials overp-adic number fields. Ann. Math. (2)56, 473–478 (1952).

    Google Scholar 

  13. Schmidt, W. M.: Simultaneousp-adic zeros of quadratic forms. Mh. Math.90, 45–65 (1980).

    Google Scholar 

  14. Schmidt, W. M.: Simultaneous rational zeros of quadratic forms. Progress Math. (In print.)

  15. Schmidt, W. M.: On cubic polynomials III. Systems ofp-adic equations. Mh. Math.93, 211–223 (1982).

    Google Scholar 

  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. 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. Watson, G. L.: Non-homogeneous cubic equations. Proc. London Math. Soc. (3)17, 271–295 (1967).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, University of Colorado, 80309, Boulder, CO, USA

    Wolfgang M. Schmidt

Authors
  1. Wolfgang M. Schmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by NSF contract NSF-MCS-8015356.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schmidt, W.M. On cubic polynomials. Monatshefte für Mathematik 93, 329–348 (1982). https://doi.org/10.1007/BF01295233

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01295233

Keywords

Search

Navigation