Solutions of full equations related to diagonal equations

TL;DR

The work counts solutions in finite fields for diagonal and related full equations by leveraging pure Gauss sums under arithmetic admissibility, deriving explicit formulas for the number of $*$-roots $N^*(g)$ of diagonal polynomials and transferring these counts to full polynomials via $*$-equivalence; the results express $N^*(g)$ in terms of $C_1(d)$, $C_2(d)$, and the Legendre-type character $\eta_d$, with $q$ required to be a square and $d$ $(p,r)$-admissible. This yields closed-form expressions for $N(f)$ when the full polynomial is $*$-equivalent to a diagonal one, including the contributions from zeros of coordinates. The paper includes concrete examples over $\mathbb{F}_{81}$ and $\mathbb{F}_{256}$ to demonstrate the method and underscores the utility of Gauss-sum techniques in counting points on diagonal and full Diophantine-type hypersurfaces over finite fields.

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.

Theorems & Definitions (21)

• Lemma 1
• proof
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6