某些对角方程在有限域上的解数

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s∑(i=1) a_ix_i~(m_i)=c的对角方程,其中a_i,c∈F_q~2~*,(m_i,m_j)=1,m_i|(q+1),m_i为奇数或(q+1)/(m_i)为偶数,i=1,2,…,s,以及形如s∑(i=1) x_i~m=c的对角方程,其中c∈F_q~*,m|(q+1),m为奇数或(q+1)/m为偶数.

The Number of Solutions of Certain Diagonal Equations over Finite Fields

In this paper, using some properties about Gaussian sums and Jacobi sums, the authors get the explicit formulasfor the number of solutions of the equation $sumlimits_{i=1}^{s}a_ix_i^{m_i}=c$,where $a_i$, $cinmathbb F_{q^2}^*$, $(m_i,m_j)=1$, $m_i|(q+1)$, $m_i$ odd or $frac{q+1}{m_i}$is even, $i=1,2,cdots,s$,and the equation $sumlimits_{i=1}^{s}x_i^{m}=c$, where $cinmathbb F^*_q$, $m|(q+1)$, $m$ odd or $frac{q+1}{m}$ is even.