Estimates for exponential sums

Abstract:

If f is a polynomial over Z of degree $n + 1$ with $n \geqslant 1$, then for each integer $q \geqslant 1,|{\Sigma _{1 \leqslant x \leqslant q}}\exp (2\pi if(x)/q)| \leqslant {q^{1/2}}(D,q){d_n}(q)$, provided the discriminant D of the derivative of f does not vanish identically, where ${d_n}(q)$ is the number of representations of q as a product of n factors.