| 顾晶晶,曹喜望.某些对角方程在有限域上的解数[J].数学年刊A辑,2018,39(2):211~218 |
| 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 |
| Received:June 18, 2015 Revised:August 17, 2017 |
| DOI：10.16205/j.cnki.cama.2018.0020 |
| 中文关键词: 对角方程, 解数, Gauss和, Jacobi和, 有限域 |
| 英文关键词:Diagonal equation, Number of solutions, Gaussian sum, Jacobi sum, Finite field |
| 基金项目:本文受到国家自然科学基金(No.11371011)的资助. |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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. |
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