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An improved Mordell type bound for exponential sums. (English) Zbl 1068.11053

The authors consider mixed exponential sums of the type \[ S(\chi ,f) = \sum_{x=1}^{p-1}\chi (x)e(f(x)/p), \] where \(p\) is a prime, \(\chi \) a multiplicative character \(\pmod{p}\), and \(f(x)=a_1x^{k_1}+ \dotsb + a_rx^{k_r}\) with \(p\nmid a_i\) and \(k_i \in \mathbb Z\). In the special case where \(\chi \) is trivial and \(1\leq k_1 < \cdots < k_r\), the bound \(2^{2/r}(k_1\dotsb k_r)^{1/r^2}p^{1-1/(2r)}\) for the modulus of the exponential sum is obtained. This improves a result of Mordell from 1932. A key ingredient of the argument is bounding the number of solutions of the system of congruences \[ \sum_{i=1}^r x_i^{k_j} \equiv \sum_{i=1}^ry_i^{k_j} \pmod{p}, \quad (x_iy_i,p)=1, \quad j=1,\dotsc , r. \]

MSC:

11L07 Estimates on exponential sums
11L05 Gauss and Kloosterman sums; generalizations
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References:

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