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Proof of a conjectured three-valued family of Weil sums of binomials

Volume 169 / 2015

Daniel J. Katz, Philippe Langevin Acta Arithmetica 169 (2015), 181-199 MSC: Primary 11T23, 11T24, 11L05; Secondary 11G20, 05C90. DOI: 10.4064/aa169-2-5

Abstract

We consider Weil sums of binomials of the form $$W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x),$$ where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we fix $F$ and $d$, and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $F$ is a field of order $3^n$ with $n$ odd, and $d=3^r+2$ with $4 r \equiv 1\ {\rm mod}\ n$, then $W_{F,d}(a)$ assumes only the three values $0$ and $\pm 3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.

Authors

  • Daniel J. KatzDepartment of Mathematics
    California State University, Northridge
    18111 Nordhoff St.
    Northridge, CA 91330-8313, U.S.A.
    e-mail
  • Philippe LangevinInstitut de Mathématiques de Toulon
    Université de Toulon
    Avenue de l'Université
    83957 La Garde Cedex, France
    e-mail

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