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

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Katz D., Langevin P.

Acta Arithmetica

2015

tom 169

nr 2

181-199

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.

Ograniczanie wyników

1 Acta Arithmetica

1 Katz D.

1 Langevin P.

1 2015

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