Solutions of full equations related to diagonal equations

Abstract

Let $p$ be a prime number, $m$ be an even positive integer, and $\mathbb{F}_q$ be a finite field with $q = p^m$ elements. In this paper, we compute the number of solutions with all coordinates in $\mathbb{F}_q^*$ for diagonal equations of the form $$a_1 x_1^{d} + \dots + a_s x_s^{d} = b, \quad a_i \in \mathbb{F}_q^*, \, b \in \mathbb{F}_q,$$ when the coefficients and exponents satisfy specific arithmetic conditions that facilitate the computation through pure Gauss sums. We then apply this result to determine the number of solutions for equations of the form $$a_1 x_1^{d_{1,1}} \cdots x_n^{d_{n,1}} + \dots + a_s x_1^{d_{1,s}}\cdots x_n^{d_{n,s}} = b,$$ where all exponents are positive, and the equation is related in a particular way to diagonal equations with the aforementioned characteristics.