Let F q stand for the finite field of odd characteristic p with q elements (q = p n , n ∈ N) and F * q denote the set of all the nonzero elements of 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 F q :where the integers t > 0, r 0 = 0 < r 1 < r 2 < ... < r t , 1 ≤ n 1 < n 2 < ... < n t , 0 ≤ j ≤ t − 1, b k ∈ F q , a k,i ∈ 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.

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Let 𝔽 q {\mathbb{F}_{q}} be the finite field of q = p m ≡ 1 ( mod 4 ) {q=p^{m}\equiv 1~{}(\bmod~{}4)} elements with p being an odd prime and m being a positive integer. For c , y ∈ 𝔽 q {c,y\in\mathbb{F}_{q}} with y ∈ 𝔽 q * {y\in\mathbb{F}_{q}^{*}} non-quartic, let N n ⁢ ( c ) {N_{n}(c)} and M n ⁢ ( y ) {M_{n}(y)} be the numbers of zeros of x 1 4 + ⋯ + x n 4 = c {x_{1}^{4}+\cdots+x_{n}^{4}=c} and x 1 4 + ⋯ + x n - 1 4 + y ⁢ x n 4 = 0 {x_{1}^{4}+\cdots+x_{n-1}^{4}+yx_{n}^{4}=0} , respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function ∑ n = 1 ∞ N n ⁢ ( 0 ) ⁢ x n {\sum_{n=1}^{\infty}N_{n}(0)x^{n}} is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions ∑ n = 1 ∞ N n ⁢ ( c ) ⁢ x n   and   ∑ n = 1 ∞ M n + 1 ⁢ ( y ) ⁢ x n \sum_{n=1}^{\infty}N_{n}(c)x^{n}\quad\text{and}\quad\sum_{n=1}^{\infty}M_{n+1}% (y)x^{n} are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.

Let $f(x)\in \mathbb {Z}[x]$ be a nonconstant polynomial. Let $n\ge 1, k\ge 2$ and c be integers. An integer a is called an f-exunit in the ring $\mathbb {Z}_n$ of residue classes modulo n if $\gcd (f(a),n)=1$ . We use the principle of cross-classification to derive an explicit formula for the number ${\mathcal N}_{k,f,c}(n)$ of solutions $(x_1,\ldots ,x_k)$ of the congruence $x_1+\cdots +x_k\equiv c\pmod n$ with all $x_i$ being f-exunits in the ring $\mathbb {Z}_n$ . This extends a recent result of Anand et al. [‘On a question of f-exunits in $\mathbb {Z}/{n\mathbb {Z}}$ ’, Arch. Math. (Basel)116 (2021), 403–409]. We derive a more explicit formula for ${\mathcal N}_{k,f,c}(n)$ when $f(x)$ is linear or quadratic.