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

• Autorzy: Daniel J. Katz , Philippe Langevin
• Języki publikacji: EN

Abstrakty

EN

We consider Weil sums of binomials of the form
$W_{F,d}(a) = ∑_{x∈ F} ψ(x^{d} - ax)$,
where F is a finite field, ψ: F → ℂ is the canonical additive character, $gcd(d,|F^{×}|) = 1$, and $a ∈ F^{×}$. If we fix F and d, and examine the values of $W_{F,d}(a)$ as a runs through $F^{×}$, we always obtain at least three distinct values unless d is degenerate (a power of the characteristic of F modulo $|F^{×}|$). 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 3ⁿ with n odd, and $d = 3^{r} + 2$ with 4r ≡ 1 (mod n), then $W_{F,d}(a)$ assumes only the three values 0 and $±3^{(n+1)/2}$. 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.

Kategorie tematyczne

• 11G20: Curves over finite and local fields
• 05C90: Applications
• 11T23: Exponential sums
• 11L05: Gauss and Kloosterman sums; generalizations
• 11T24: Other character sums and Gauss sums

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 2015
• Tom: 169
• Numer: 2
• Strony: 181-199

Daty

wydano

2015

Twórcy

autor

Daniel J. Katz

• Department of Mathematics, California State University, Northridge, 18111 Nordhoff St., Northridge, CA 91330-8313, U.S.A.

autor

Philippe Langevin

• Institut de Mathématiques de Toulon, Université de Toulon, Avenue de l'Université, 83957 La Garde Cedex, France

Identyfikatory

DOI

10.4064/aa169-2-5