  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.
