Theorem 3 Let $k$ be an integer $\geqslant 3$ and $q$ be a positive integer. Let
$$f(x)=a_{k} x^{k}+a_{k-1} x^{k-1}+\cdots+a_{1} x+a_{0}$$

where $a_{i}(i=0,1,2, \cdots, k)$ are all integers. When $\left(a_{k}, a_{k-1}, \cdots \cdots\right.$, $\left.a_{1}, q\right)=1$, we have
$$\left|\sum_{x=1}^{q} e^{2 \pi i \frac{(x)}{q}}\right| \leqslant C_{2}(k) q^{1-\frac{1}{k}}$$

where
$$C_{2}(k)=\left\{\begin{array}{cc} e^{4 k}, & \text { when } k \geqslant 10 \text {; } \\ e^{C_{3}(k) k}, & \text { when } 3 \leqslant k \leqslant 9 \text {; } \end{array}\right.$$

and $C_{3}(3)=6.1, C_{3}(4)=5.5, C_{3}(5)=5, C_{3}(6)=4.7$,
$$C_{3}(7)=4.4, \quad C_{3}(8)=4.2, \quad C_{3}(9)=4.05$$