Monatshefte für Mathematik

, Volume 93, Issue 1, pp 63–74

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

  • Wolfgang M. Schmidt
Article

DOI: 10.1007/BF01579030

Cite this article as:
Schmidt, W.M. Monatshefte für Mathematik (1982) 93: 63. doi:10.1007/BF01579030

Abstract

Hua andChen gave estimates of sums\(\sum\limits_{x = 1}^q {e(\mathfrak{F}(x))} \) wheree(z)=e2πiz and\(\mathfrak{F}\) is a polynomial of the typef(x)/q wheref(x)=akxk+...+a1x with integer coefficients having gcd (q, ak,...,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.

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

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