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

Daniel J. Katz¹ ; Philippe Langevin²

¹ Department of Mathematics California State University, Northridge 18111 Nordhoff St. Northridge, CA 91330-8313, U.S.A.
² Institut de Mathématiques de Toulon Université de Toulon Avenue de l'Université 83957 La Garde Cedex, France

Acta Arithmetica, Tome 169 (2015) no. 2, pp. 181-199

Résumé

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.

Détail

DOI : 10.4064/aa169-2-5

Keywords: consider weil sums binomials form sum psi d a where finite field psi colon mathbb canonical additive character gcd times times fix examine values runs through times always obtain least three distinct values unless degenerate power characteristic modulo nbsp times choices which obtain only three values quite rare desirable wide variety applications field order odd equiv mod assumes only three values proves conjecture dobbertin helleseth kumar martinsen proof employs diverse methods involving trilinear forms counting points curves via multiplicative character sums divisibility properties gauss sums graph theory