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

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.