Counting rational points of an algebraic variety over finite fields

Abstract

Let $\mathbb{F}_q$ denote the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ represent the nonzero elements of $\mathbb{F}_{q}$. In this paper, by using the Smith normal form we give an explicit formula for the number of rational points of the algebraic variety defined by the following system of equations over $\mathbb{F}_{q}$: \begin{align*} {\left\{\begin{array}{rl} &\sum_{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_1}^{e^{(1)}_{i,n_1}} +\sum_{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} ...x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ &\sum_{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_3}^{e^{(2)}_{j,n_3}} +\sum_{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} ...x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right.} \end{align*} where the integers $1\leq r_1<r_2$, $1\leq r_3<r_4$, $1\le n_1<n_2$, $1\le n_3<n_4$, $n_1\leq n_3$, $b_1, b_2\in \mathbb{F}_{q}$, $a_{1i}\in \mathbb{F}_{q}^{*}$ $(1\leq i\leq r_2)$, $a_{2j}\in \mathbb{F}_{q}^{*}$$(1\leq j\leq r_4)$ and the exponent of each variable is a positive integer. An example is also presented to demonstrate the validity of the main result.