# zbMATH — the first resource for mathematics

Further estimate of complete trigonometric sums. (English) Zbl 0934.11038
Summary: The following theorem is proved: Let $$f(x)= a_k z^k+\cdots+ a_1x+ a_0$$ be a polynomial with integral coefficients such that $$(a_1,\dots, a_k,q)=1$$, where $$q$$ is a positive integer. Then, for $$k\geq 3$$, $\Biggl|\sum_{x=1}^q e^{2\pi if(x)/q} \Biggr|\leq e^{1.74k} q^{1-\frac 1k}.$ This improved earlier estimates of Nechaev, Chen Jingrun, Stechkin, Lu Minggao and the authors.

##### MSC:
 11L07 Estimates on exponential sums 11L03 Trigonometric and exponential sums (general theory)