LAUSR

Abstract Let 𝔽 q {\mathbb{F}_{q}} be the finite field of odd characteristic p with q elements ( q = p n {q=p^{n}} , n ∈ ℕ {n\in\mathbb{N}} ) and let 𝔽 q * {\mathbb{F}_{q}^{*}} represent the set of nonzero elements of 𝔽 q {\mathbb{F}_{q}} . By making use of the Smith normal form of exponent matrices, we obtain an explicit formula for the number of rational points on the variety defined by the following system of equations over 𝔽 q {\mathbb{F}_{q}} : { ∑ i = 1 r a i ( 1 ) ⁢ x 1 e i ⁢ 1 ( 1 ) ⁢ ⋯ ⁢ x n e i ⁢ n ( 1 ) = b 1 , ∑ j ′ = 0 t - 1 ∑ i ′ = 1 r j ′ + 1 - r j ′ a r j ′ + i ′ ( 2 ) ⁢ x 1 e r j ′ + i ′ , 1 ( 2 ) ⁢ ⋯ ⁢ x n j ′ + 1 e r j ′ + i ′ , n j ′ + 1 ( 2 ) = b 2 , \left\{\begin{aligned} &\displaystyle\sum_{i=1}^{r}a^{(1)}_{i}x_{1}^{e_{i1}^{(% 1)}}\cdots x_{n}^{e_{in}^{(1)}}=b_{1},\\ &\displaystyle\sum^{t-1}_{j^{\prime}=0}\sum^{r_{j^{\prime}+1}-r_{j^{\prime}}}_% {i^{\prime}=1}a^{(2)}_{r_{j^{\prime}}+i^{\prime}}x_{1}^{e_{r_{j^{\prime}}+i^{% \prime},1}^{(2)}}\cdots x_{n_{{j^{\prime}}+1}}^{e_{r_{j^{\prime}}+i^{\prime},n% _{{j^{\prime}}+1}}^{(2)}}=b_{2},\end{aligned}\right.\vspace*{1mm} where b i ∈ 𝔽 q {b_{i}\in\mathbb{F}_{q}} ( i = 1 , 2 {i=1,2} ), t ∈ ℕ {t\in\mathbb{N}} , 0 = n 0 < n 1 < n 2 < ⋯ < n t , 0=n_{0}