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

Abstract

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 $$M_f(x):=\prod _{\genfrac{}{}{0ex}{}{m\in {\mathbb{F}}_q^{\ast }}{S_{f,m}\ne 0}}(x-\frac{1}{S_{f,m}})$$ 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 $\mathrm{Tr}$ 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.

13 pages

References in corpus (1)

• On the number of zeros of diagonal cubic forms over finite fields