On exponential sums with sparse polynomials and rational functions.(English)Zbl 0881.11081

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: $S(f,q)= O(q^{1-(1/t)+ \varepsilon}) \qquad\text{and}\qquad T_m(f)\leq r2^{m/2}.$

MSC:

 11T23 Exponential sums 11T24 Other character sums and Gauss sums 11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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