On a class of diagonal equations over finite fields

TL;DR

This paper derives explicit formulas for counting solutions to specific diagonal equations over finite fields with characteristic conditions, utilizing properties of Gauss and Jacobi sums, and expressing results through quadratic partitions of powers of p.

Contribution

It provides new explicit solution counts for diagonal equations over finite fields with characteristic p ≡ ±3 mod 8, using advanced number theoretic sums.

Findings

Explicit formulas for solution counts derived

Results expressed via quadratic partitions of p

Applicable to finite fields with specific characteristics

Abstract

Using properties of Gauss and Jacobi sums, we derive explicit formulas for the number of solutions to a diagonal equation of the form $x_1^{2^m}+\dots+x_n^{2^m}=0$ over a finite field of characteristic $p\equiv\pm 3\pmod{8}$. All of the evaluations are effected in terms of parameters occurring in quadratic partitions of some powers of $p$.