On an estimate of complete trigonometric sums (Q795091)

Let q be an integer \(>1\) and \(f(x)=a_ kx^ k+...+a_ 1x+a_ 0\) be a polynomial of degree k with integral coefficients such that \((a_ 1,...,a_ k,q)=1\). By a complete trigonometric sum we mean a sum of the form \(S(q,f(x))=\sum^{q}_{x=1}e^{2\pi i f(x)/q}.\) Let \(s(q,f)=| S(q,f(x))| /q^{1-1/k}.\) In 1940 \textit{L. Hua} [C. R. Acad. Sci., Paris 210, 520-523 (1940; Zbl 0023.01101)] first proved that \(s(q,f)=O(q^{\epsilon})\) where the constant implied by ''O'' depends only on \(k\) and \(\epsilon\). The main order \(1-1/k\) is the best possible. Later on the following improvements were made. In 1977, \textit{J. Chen} [Sci. Sin. 20, 711-719 (1977; Zbl 0374.10024)] and \textit{S. B. Stechkin} [Tr. Mat. Inst. Steklova 143, 188-207 (1977; Zbl 0433.10026)] proved, respectively, \(s(q,f)\leq e^{6.1 k}\) (\(k\geq 3)\) and \(s(q,f)\leq \exp(k+O(k/\log k))\). In this paper \(s(q,f)\leq e^{2k}\) is proved.