Divisibility of Weil Sums of Binomials

TL;DR

This paper investigates the divisibility properties of Weil sums of binomials over finite fields, establishing bounds and proving a conjecture for characteristic 3 fields related to the rarity of three-valued sums.

Contribution

It provides a lower bound on the p-adic valuation of three-valued Weil sums and proves a special case of Helleseth's conjecture for characteristic 3 fields.

Findings

Lower bound on p-adic valuation of three-valued Weil sums

Proved Helleseth's conjecture for characteristic 3 and degree a power of 2

Three-valued Weil sums are rare and constrained by divisibility properties

Abstract

Consider the Weil sum $W_{F,d}(u)=\sum_{x \in F} \psi(x^d+u x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, $d$ is coprime to $|F^*|$, and $u \in F^*$. We say that $W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as $u$ runs through $F^*$: this is the minimum number of distinct values in the nondegenerate case, and three-valued $W_{F,d}$ are rare and desirable. When $W_{F,d}$ is three-valued, we give a lower bound on the $p$-adic valuation of the values. This enables us to prove the characteristic $3$ case of a 1976 conjecture of Helleseth: when $p=3$ and $[F:{\mathbb F}_3]$ is a power of $2$, we show that $W_{F,d}$ cannot be three-valued.