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

## Authors

- Received:

DOI: 10.1007/BF01579030

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

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## Abstract

Hua andChen gave estimates of sums\(\sum\limits_{x = 1}^q {e(\mathfrak{F}(x))} \) where*e(z)=e*^{2πiz} and\(\mathfrak{F}\) is a polynomial of the type*f(x)/q* where*f(x)=a*_{k}*x*^{k}+...+*a*_{1}*x* with integer coefficients having gcd (*q, a*_{k},...,*a*_{1})=1 But no good estimates hold for these sums when*q* is small in comparison to*k*. 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 with*p*-adic and rational zeros of systems of cubic forms.