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 a formula for the number of rational points of the algebraic varieties defined by the following system of equations over ${\mathbb {F}}_{q}$:

$$\begin{aligned} \left\{ \begin{array}{ll} \sum \limits _{i=1}^{r_1}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_1}^{e^{(1)}_{i,n_1}} +\sum \limits _{i=r_1+1}^{r_2}a_{1i}x_1^{e^{(1)}_{i1}} \ldots x_{n_2}^{e^{(1)}_{i,n_2}}-b_1=0,\\ \sum \limits _{j=1}^{r_3}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_3}^{e^{(2)}_{j,n_3}} +\sum \limits _{j=r_3+1}^{r_4}a_{2j}x_1^{e^{(2)}_{j1}} \ldots x_{n_4}^{e^{(2)}_{j,n_4}}-b_2=0, \end{array}\right. \end{aligned}$$

where the integers $1\le r_1<r_2$, $1\le r_3<r_4$, $1\le n_1<n_2$, $1\le n_3<n_4$, $n_1\le n_3$, $b_1, b_2\in \mathbb {F}_{q}$, $a_{1i}\in \mathbb {F}_{q}^{*}(1\le i\le r_2)$, $a_{2j}\in \mathbb {F}_{q}^{*}(1\le j\le r_4)$ and the exponent of each variable is positive integer. Our result provides a partial answer to an open problem raised in Hu et al. (J Number Theory 156:135–153, 2015).