有限域上一类四次对角方程有理点的个数
The number of solutions of certain quartic diagonal equations over finite field
胡双年 1高继东 2杜屹洋1
作者信息
- 1. 南阳理工学院数理学院, 南阳 473004
- 2. 南阳理工学院图书馆, 南阳 473004
- 折叠
摘要
设p为素数,k为正整数,Fq是 q= pk的有限域.用F*q表示Fq的乘法群,即F*q=Fq\{0}.设f(x1,…,xn)是Fq上的多项式,用 N(f(x1,…,xn)=0)表示 f(x1,x2,…,xn)=0在Fq上的有理点个数.1981年,Myerson 给出了N(x41 +⋯+ x4n=0)的递推公式.最近,赵等给出了 N(x41 + x42 = c),N(x41 + x42 + x43 = c)和 N(x41 + x42 + x43 + x44 = c)的精确公式,其中c∈ F*q.本文利用雅可比和以及一个类比Hasse-Davenport定理的结果给出了N(x41 +⋯+ x4n= c)的精确公式,扩展了已有结果.
Abstract
Let p be a prime,k be a positive integer and Fq be the finite field of q= pk elements.Let F*q be the multiplicative group of Fq,that is,F*q=Fq\{0}.For a polynomial f(x1,…,xn)over Fq,use N(f(x1,…,xn)=0)to denote the number of solutions of f(x1,x2,…,xn)=0 over Fq.In 1981,Myerson gave a formula for N(x41 +⋯+ x4n=0).Recently,Zhao and coworkers obtained the explicit formulas for N(x41 + x42 = c),N(x41 + x42 + x43 = c)and N(x41 + x42 + x43 + x44 = c),where c∈ F*q.In this paper,by using the Jacobi sums and an analog of the Hasse-Davenport theorem,we obtain the exact formula for N(x41 +⋯+ x4n= c)and thus extend the known results.
关键词
有限域/有理点/对角方程/雅可比和Key words
Finite field/Rational point/Diagonal equation/Jacobi sum引用本文复制引用
基金项目
国家自然科学基金(12026224)
河南省自然科学基金(232300420123)
出版年
2024
国家科技期刊平台
NSTL
万方数据