Monatshefte für Mathematik

, Volume 93, Issue 1, pp 63–74 | Cite as

On cubic polynomials I. Hua's estimate of exponential sums

  • Wolfgang M. Schmidt


Hua andChen gave estimates of sums\(\sum\limits_{x = 1}^q {e(\mathfrak{F}(x))} \) wheree(z)=e 2πiz and\(\mathfrak{F}\) is a polynomial of the typef(x)/q wheref(x)=a k x k +...+a1x with integer coefficients having gcd (q, a k ,...,a1)=1 But no good estimates hold for these sums whenq is small in comparison tok. We therefore consider here a related but different class of polynomials. Special emphasis is given to the cubic case.

In subsequent papers of this series we shall deal with cubic exponential sums in many variables and withp-adic and rational zeros of systems of cubic forms.


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  1. [1]
    Chen, Jing R.: On Professor Hua's estimate of exponential sums. Sci. Sinica20, 711–719 (1977).Google Scholar
  2. [2]
    Hua, L. K.: Additive prime number theory. (Chinese). Peking: Science Press. 1957.Google Scholar
  3. [3]
    Schmidt, W. M.: Simultaneousp-adic zeros of quadratic forms. Mh. Math.90, 45–65 (1980).Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

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

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