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Pure and mixed exponential sums. (English) Zbl 0937.11031
The authors consider bounds for the exponential sums $\text{(A)}\quad S(f,p^m)= \sum_{x=1}^{p^m} e_{p^m} (f(x)), \qquad \text{(B)}\quad S(\chi,f,p^m)= \sum _{\substack{ x=1,\\ p\nmid x}}^{p^m} \chi(x) e_{p^m} (f(x)),$ where $$p^m$$ is a prime power with $$m\geq 2$$, $$\chi$$ is a multiplicative character modulo $$p^m$$, $$e_{p^m}(x)= e^{2\pi ix/p^m}$$ and $$f$$ is a polynomial with integer coefficients and degree $$d>1$$. For (A), the critical points are defined by $$p^{-t} f'(x)\equiv 0\bmod p$$, where $$t= \operatorname {ord}_p (f')$$. A consequence of the more precise estimates in the paper is $|S(f,p^m)|\leq (d-1) p^{t/(M+1)} p^{m(1-1/(M+1))},$ providing $$m\geq t+2$$, where $$M$$ is the maximum multiplicity of the critical points. The estimates are obtained by a refinement of Chalk’s approach involving reduction of the sum locally around each critical point. Similar results have been obtained recently by P. Ding [J. Number Theory 65, 116-129 (1997; Zbl 0876.11044)] and W. K. A. Loh [Bull. Aust. Math. Soc. 50, 451-458 (1994; Zbl 0833.11040)]. Bounds for (B) follow by a similar technique by treating the sum as a twisted version of (A). The character $$\chi$$ can be defined by $$\chi(a^k)= e(ck/p^{m-1} (p-1))$$, where $$a$$ is a fixed primitive root modulo $$p$$ chosen so that $$a^{p-1}= 1+rp$$ with $$p\nmid r$$ and $$c$$ is uniquely determined by $$\chi$$ and $$a$$ and $$0\leq c\leq p^{m-1} (p-1)$$. Let $$t= \operatorname {ord}_p (f'(x))$$, $$t_1= \operatorname {ord}_p (rxf'(x)+c)$$. The critical points are defined by $$p^{-t_1} (rxf'(x)+c)\equiv 0\bmod p$$. A consequence of much more precise and detailed estimates is $|S(\chi,f,p^m)|\leq dp^{t/(M+1)} p^{m(1-1/(M+1))},$ providing $$m\geq t_1+2$$, where $$M$$ is the maximum multiplicity of the critical points.

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