Diagonal equations with restricted solution sets

TL;DR

This paper derives explicit formulas for counting solutions to diagonal equations over finite fields with variables restricted to subfields, using quadratic form theory, and discusses conditions for solution existence and open problems.

Contribution

It introduces a method to explicitly count solutions of diagonal equations with restricted variables over finite fields, extending previous results with new formulas and existence conditions.

Findings

Explicit solution count formulas for diagonal equations with restricted variables.

Conditions for the existence of solutions based on exponents and field restrictions.

Discussion of open problems and future research directions.

Abstract

Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ with $x_i\in\mathbb{F}_{p^{t_i}}$, where $b\in\mathbb{F}_q$ and $t_i|n$ for all $i=1,\dots,s$. In our main results, we employ results on quadratic forms to give an explicit formula for the number of solutions of diagonal equations with restricted solution sets satisfying certain natural restrictions on the exponents. As a consequence, we present conditions for the existence of solutions. We also discuss further questions concerning equations with restricted solution sets and present some open problems.