On exponential sums with sparse polynomials and rational functions (Q1126370)

Let \(f(x)= a_1x^{r_1}+\cdots+ a_tx^{r_t}\), where \(r_1,\dots, r_t\) are some pairwise distinct nonzero integers, be a sparse polynomial with integral coefficients. Suppose that \[ S(f,q)=\sum^q_{\substack{ x=1\\ (x,q)=1}} \exp[2\pi if(x)/q], \] where \(q\) is an integer and \((a_1,\dots, a_t,q)=1\), and \(T_m(f)= \sum_{x\in\mathbb{F}_{2^m}} \chi(f(x))\), where \(\chi\) is a nontrivial additive character of \(\mathbb{F}_{2^m}\). In the present paper, the author proves the following: \[ S(f,q)= O(q^{1-(1/t)+ \varepsilon}) \qquad\text{and}\qquad T_m(f)\leq r2^{m/2}. \]