Solutions of full equations related to diagonal equations

2024/03/19 by Coelho, José Gustavo
#11D45 #11T06 #11T24 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2403.13133

Abstract

Let p be a prime number, m be an even positive integer, and \mathbbF_q be a finite field with q = p^m elements. In this paper, we compute the number of solutions with all coordinates in \mathbbF_q^* for diagonal equations of the form a₁ x₁^d + … + a_s x_s^d = b, a_i ∈ \mathbbF_q^*, b ∈ \mathbbF_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 a₁ x₁^d_1,1 ⋯ x_n^d_n,1 + … + a_s x₁^d_1,s⋯ 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.

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