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

Daniel J. KatzPhilippe Langevin — 2015

Acta Arithmetica

We consider Weil sums of binomials of the form W F , d ( a ) = x F ψ ( x d - a x ) , where F is a finite field, ψ: F → ℂ is the canonical additive character, g c d ( 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...

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