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Estimate of a complete trigonometric sum. (English) Zbl 0599.10030
The paper concerns the estimation of $S(q,f(x))=\sum^{q}_{x=1}e^{2\pi if(x)/q}$ where $$f(x)=a_ kx^ k+...+a_ 1x+a_ 0$$ and satisfies $$k\geq 3$$ and $$(a_ 1,...,a_ k,q)=1.$$ Many estimates have been given of the form $| S(q,f(x))| \quad \leq \quad C(k) q^{1-1/k}$ and in this paper the author obtains improved values for $$C(k)$$. In the case that $$q$$ is a prime power his estimates together imply that $$C(k)\leq k$$. For general $$q$$ the estimate $$C(k)\leq e^{1.85k}$$ is obtained and it is stated that this can be improved to $$C(k)\leq e^{1.76k}$$ for $$k\geq 12.$$
The proof is based on a more detailed study of polynomial congruences modulo a power of a prime.
Reviewer: D.A.Burgess

##### MSC:
 11L40 Estimates on character sums