Solutions of equations over finite fields: enumeration via bijections

TL;DR

This paper provides elementary combinatorial proofs for the number of solutions to diagonal and Carlitz equations over finite fields, avoiding character sums and simplifying existing proofs.

Contribution

It introduces a new combinatorial approach to count solutions of certain equations over finite fields, simplifying and generalizing previous results.

Findings

Elementary combinatorial proofs for solution counts

Simplification of existing proofs without character sums

Generalization of solutions for Carlitz equations

Abstract

We present a simple proof of the well-known fact concerning the number of solutions of diagonal equations over finite fields. In a similar manner, we give an alternative proof of the recent result on generalizations of Carlitz equations. In both cases, the use of character sums is avoided by using an elementary combinatorial argument.