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Upper bounds on a two-term exponential sum. (English) Zbl 1012.11078
The authors derive upper bounds for mixed exponential sums of type $S( \chi, a x^n + bx, p^m) = \sum_{_{\substack{ x=1\\ p \nmid x}}}^{p^m} \chi(x) e_{p^m}(a x^n + bx),$ where $$p^m$$ is a prime power with $$m \geq 2$$, $$a,b$$ are integers, $$n \geq 2$$, $$\chi$$ is a multiplicative character mod $$p^m$$ and $$e_{p^m}(x) = \exp(2 \pi i x/p^m)$$ . If $$\chi$$ is primitive or $$p \nmid (a,b)$$ , then they obtain $|S(\chi, ax^n +bx, p^m)|\leq 2 n p^{2m/3}.$ If $$\chi$$ is of conductor $$p$$ with $$p \nmid (a,b)$$, then $|S(\chi, ax^n +bx, p^m)|\leq n p^{m/2}.$

##### MSC:
 11L07 Estimates on exponential sums
##### Keywords:
upper bounds; mixed exponential sums
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##### References:
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