  Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be the
finite field of $q=p^k$ elements. Let $f(x)$ be a polynomial over $\mathbb F_q$
and $a\in\mathbb F_q$. We denote by $N_{s}(f,a)$ the number of zeros of
$f(x_1)+\cdots+f(x_s)=a$. In this paper, we show that
$$\sum_{s=1}^{\infty}N_{s}(f,0)x^s=\frac{x}{1-qx} -\frac{x {
M_f^{\prime}}(x)}{qM_f(x)},$$ where $$M_f(x):=\prod_{m\in\mathbb
F_q^{\ast}\atop{S_{f, m}\ne 0}}\Big(x-\frac{1}{S_{f,m}}\Big)$$ with $S_{f,
m}:=\sum_{x\in \mathbb F_q}\zeta_p^{{\rm Tr}(mf(x))}$, $\zeta_p$ being the
$p$-th primitive unit root and ${\rm Tr}$ being the trace map from $\mathbb
F_q$ to $\mathbb F_p$. This extends Richman's theorem which treats the case of
$f(x)$ being a monomial. Moreover, we show that the generating series
$\sum_{s=1}^{\infty}N_{s}(f,a)x^s$ is a rational function in $x$ and also
present its explicit expression in terms of the first $2d+1$ initial values
$N_{1}(f,a), ..., N_{2d+1}(f,a)$, where $d$ is a positive integer no more than
$q-1$. From this result, the theorems of Chowla-Cowles-Cowles and of Myerson
can be derived.
