On the Number of Solutions to the Equation (x₁ + ⋯ + x _n )^m = ax₁ ⋯ x _n over the Finite Field $\mathbb{F}$ _q for gcd(m − n, q − 1) = 7 and gcd(m − n, q − 1) = 14

Abstract

Let N _q be the number of solutions to the equation

$${\left( {{x_i} + \cdots + {x_n}} \right)^m} = a{x_1} \cdots {x_n}$$

over the finite field $\mathbb{F}$ _q = $\mathbb{F}$ _p s . L. Carlitz found formulas for N _q for m = 2, n = 3 or 4, and p < 2. In earlier papers, we found formulas for N _q when d = gcd(m − n, q − 1) = 1 or 2 or 3 or 4 or 6 or 8; and when there exists an ℓ such that d|(p^ℓ + 1). In this paper, we find formulas for N _q when d = 7 or 14 and p ≡ 2 or 4 (mod 7).