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Let $p$ be a prime, $k$ a positive integer and let $\mathbb{F}_q$ be thefinite 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 $$M_f(x):=\prod_{m\in\mathbbF_q^{\ast}\atop{S_{f, m}\ne 0}}\Big(x-\frac{1}{S_{f,m}}\Big)$$ 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 $\mathbbF_q$ 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 alsopresent 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 Myersoncan be derived.

On the number of zeros to the equation $f(x_1)+...+f(x_n)=a$ over finite fields