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Divisibility of Weil sums of binomials


Author: Daniel J. Katz
Journal: Proc. Amer. Math. Soc. 143 (2015), 4623-4632
MSC (2010): Primary 11T23, 11L05, 11L07; Secondary 11T71
Published electronically: April 1, 2015
MathSciNet review: 3391022
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Abstract: Consider the Weil sum $ W_{F,d}(u)=\sum _{x \in F} \psi (x^d+u x)$, where $ F$ is a finite field of characteristic $ p$, $ \psi $ is the canonical additive character of $ F$, $ d$ is coprime to $ \vert F^*\vert$, and $ u \in F^*$. We say that $ W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as $ u$ runs through $ F^*$: this is the minimum number of distinct values in the nondegenerate case, and three-valued $ W_{F,d}$ are rare and desirable. When $ W_{F,d}$ is three-valued, we give a lower bound on the $ p$-adic valuation of the values. This enables us to prove the characteristic $ 3$ case of a 1976 conjecture of Helleseth: when $ p=3$ and $ [F:{\mathbb{F}}_3]$ is a power of $ 2$, we show that $ W_{F,d}$ cannot be three-valued.


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Additional Information

Daniel J. Katz
Affiliation: Department of Mathematics, California State University, Northridge, California 91330-8313

DOI: https://doi.org/10.1090/proc/12687
Keywords: Weil sum, character sum, finite field
Received by editor(s): July 29, 2014
Published electronically: April 1, 2015
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2015 American Mathematical Society