The number of rational points on a family of varieties over finite fields

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive integers. In this paper, by using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over $\mathbb{F}_{q}$: $$ \sum\limits_{j=0}^{t-1}\sum\limits_{i=1}^{r_{j+1}-r_j} a_{k,r_j+i}x_1^{e^{(k)}_{r_j+i,1}}...x_{n_{j+1}}^{e^{(k)}_{r_j+i,n_{j+1}}}=b_k, \ k=1,...,m. $$ where the integers $t>0$, $r_0=0<r_1<r_2<...<r_t$, $1\le n_1<n_2<...<n_t$, $0\leq j\leq t-1$, $b_k\in \mathbb{F}_{q}$, $a_{k,i}\in \mathbb{F}_{q}^{*}$, $(k=1,...,m, i=1,...,r_t)$, and the exponent of each variable is a positive integer. Furthermore, under some natural conditions, we arrive at an explicit formula for the number of the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Song, Chen, Hong, Hu and Zhao et al. Our result also answers completely an open problem raised by Song and Chen.