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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

An improved Mordell type bound for exponential sums


Authors: Todd Cochrane and Christopher Pinner
Journal: Proc. Amer. Math. Soc. 133 (2005), 313-320
MSC (2000): Primary 11L07, 11L03
DOI: https://doi.org/10.1090/S0002-9939-04-07726-3
Published electronically: September 2, 2004
MathSciNet review: 2093050
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Abstract | References | Similar Articles | Additional Information

Abstract: For a sparse polynomial $f(x)=\sum_{i=1}^r a_ix^{k_i}\in \mathbb Z [x]$, with $p\nmid a_i$ and $1\leq k_1<\cdots <k_r<p-1$, we show that

\begin{displaymath}\left\vert\sum_{x=1}^{p-1} e^{2\pi i f(x)/p} \right\vert \leq... ...rac{2}{r}} (k_1\cdots k_r)^{\frac{1}{r^2}}p^{1-\frac{1}{2r}}, \end{displaymath}

thus improving upon a bound of Mordell. Analogous results are obtained for Laurent polynomials and for mixed exponential sums.


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Additional Information

Todd Cochrane
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: cochrane@math.ksu.edu

Christopher Pinner
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506
Email: pinner@math.ksu.edu

DOI: https://doi.org/10.1090/S0002-9939-04-07726-3
Keywords: Exponential sums
Received by editor(s): July 23, 2002
Received by editor(s) in revised form: September 6, 2002
Published electronically: September 2, 2004
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.