某些对角方程在有限域上的解数
The Number of Solutions of Certain Diagonal Equations over Finite Fields

主要运用Gauss和以及Jacobi和的相关性质给出两类对角方程在有限域上的解数公式, 分别是形如$\sum\limits_{i=1}^{s}a_ix^{m_i}_i=c$的对角方程, 其中$a_i$, $c\in\mathbb F_{q^2}^*$, $(m_i,m_j)=1$, $m_i|(q+1)$, $m_i$为奇数或$\frac{q+1}{m_i}$为偶数, $i=1,2,\cdots, s$, 以及形如$\sum\limits_{i=1}^{s}x^m_i=c$的对角方程, 其中$c\in\mathbb F_q^*$, $m|(q+1)$, $m$ 为奇数或$\frac{q+1}{m}$ 为偶数.
In this paper, using some properties about Gaussian sums and Jacobi sums, the authors get the explicit formulas for the number of solutions of the equation $\sum\limits_{i=1}^{s}a_ix_i^{m_i}=c$, where $a_i$, $c\in\mathbb F_{q^2}^*$, $(m_i,m_j)=1$, $m_i|(q+1)$, $m_i$ odd or $\frac{q+1}{m_i}$is even, $i=1,2,\cdots,s$, and the equation $\sum\limits_{i=1}^{s}x_i^{m}=c$, where $c\in\mathbb F^*_q$, $m|(q+1)$, $m$ odd or $\frac{q+1}{m}$ is even.