On the number of zeros of diagonal quartic forms over finite fields

TL;DR

This paper extends Myerson's 1979 results by proving that generating functions for the number of zeros of certain diagonal quartic forms over finite fields are rational and provides explicit formulas using cyclotomic and exponential sum techniques.

Contribution

The paper introduces a new approach using cyclotomic theory to establish rationality and explicit formulas for generating functions related to zeros of diagonal quartic forms over finite fields.

Findings

Generated functions are rational in x

Explicit formulas for generating functions are obtained

Extends previous results by Myerson in 1979

Abstract

Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions $\sum_{n=1}^{\infty}N_n(c)x^n$ and $\sum_{n=1}^{\infty}M_{n+1}(y)x^n$ are rational functions in $x$. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979.