Year: 2021
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Oliveira, José Alves
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3131757207