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

Abstract

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+...+a ₁ x with integer coefficients having gcd (q, a _k ,...,a ₁)=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.