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$\sum _{s=1}^{\infty }{N}_s(f,0)x^s=\frac{x}{1-qx}-\frac{x{M}_f^{\prime }(x)}{q{M}_f(x)},$ where $M_f^{\prime }(x)$ stands for the derivative of $M_f(x)$ and$M_f(x):=\prod _{\genfrac{}{}{0ex}{}{m\in {\mathbb{F}}_q^{⁎}}{S_{f,m}\ne 0}}(x-\frac{1}{S_{f,m}})$ with $S_{f,m}:={\sum }_{x\in {\mathbb{F}}_q}{\zeta }_p^{\mathrm{Tr}(mf(x))}$, ${\zeta }_p$ being the p-th primitive unit root and 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.

设p为素数，k为正整数，令${\mathbb{F}}_q$ 是的有限域 $q={磷}^{克}$元素。让$F(X)$ 是多项式 ${\mathbb{F}}_q$ 和 $\mathrm{一种}\in {\mathbb{F}}_q$. 我们表示为$N_{秒}(F,\mathrm{一种})$ 零的数量 $F(X_1)+\cdots +F(X_{秒})=\mathrm{一种}$. 在本文中，我们表明$\sum _{秒=1}^{\infty }{N}_{秒}(F,0)X^{秒}=\frac{X}{1-qX}-\frac{X{米}_F^{\prime }(X)}{q{米}_F(X)},$ 在哪里 ${米}_F^{\prime }(X)$ 代表的导数 ${米}_F(X)$ 和${米}_F(X)：=\prod _{\genfrac{}{}{0ex}{}{米\in {\mathbb{F}}_q^{⁎}}{{秒}_{F,米}\ne 0}}(X-\frac{1}{{秒}_{F,米}})$ 和 ${秒}_{F,米}：={\sum }_{X\in {\mathbb{F}}_q}{\zeta }_{磷}^{\mathrm{时间}(米F(X))}$, ${\zeta }_{磷}$是第p个原始单位根，Tr 是来自${\mathbb{F}}_q$ 到 ${\mathbb{F}}_{磷}$. 这扩展了 Richman 定理，该定理处理了$F(X)$是单项式。此外，我们证明了生成序列${\sum }_{秒=1}^{\infty }{N}_{秒}(F,\mathrm{一种})X^{秒}$是x 中的一个有理函数，并且还根据第一个$2d+1$ 初始值 $N_1(F,\mathrm{一种}),...,N_{2d+1}(F,\mathrm{一种})$，其中d是一个不超过的正整数$q-1$. 从这个结果，可以推导出 Chowla-Cowles-Cowles 和 Myerson 的定理。