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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

11L40 Estimates on character sums