Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials

TL;DR

This paper proves a conjecture about Weil sums of binomials over finite fields, showing that under specific conditions, these sums take only three distinct values, which is rare and useful for applications.

Contribution

It proves a 2001 conjecture that certain Weil sums of binomials over finite fields of order 3^n with odd n take exactly three values, using diverse mathematical methods.

Findings

Weil sums of the form W_{F,d}(a) take only three values under specified conditions.

The result applies to finite fields of order 3^n with odd n and specific d values.

The proof confirms the rarity and structure of these three-valued Weil sums.

Abstract

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 $4 r \equiv 1 \pmod{n}$, 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.