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

Abstract

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 3 n 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.