有限域上对角三次方程解的个数  

Number of Solutions of Diagonal Cubic Equations over Finite Fields

作　　者：梁雪灵 戈文旭 LIANG Xue-ling;GE Wen-xu(School of Mathematics and Statistics,North China University of Water Resources and Electric Power,Zhengzhou 450046,China)

机构地区：[1]华北水利水电大学数学与统计学院,河南郑州450046

出　　处：《兰州文理学院学报（自然科学版）》2024年第2期20-24,共5页Journal of Lanzhou University of Arts and Science(Natural Sciences)

基　　金：国家自然科学基金项目(12071132);河南省自然科学基金项目(222300420493)。

摘　　要：利用Gauss和、指数和的性质,将求解方程解的个数问题转化为Gauss和问题.研究了在一个由q=p k(p为素数,k为正整数)元素构成的有限域F q上的对角三次方程x^(3)_(1)+x^(3)_(2)+…+x_(n)^(3)+y^(3)_(1)+y^(3)_(2)+…+y^(3)_(m)=0解的个数问题.令此方程解的个数为A n,m,对于任意的变量x∈F_(q),y∈F_(q)^(*)=(i=1,2,…,n;j=1,2,…,m,得到了生成函数∑∞m=1∑∞n=1 A(n,m)x^(n)y^(m)的显式表达式.In this paper,the numerical problems of solving equations are transformed into Gauss sum problems by using the properties of Gauss sum and exponential sum.On a finite field consisting of q=p k(p is a prime number,k is a positive integer)elements,the problem of the number of solutions to the diagonal cubic equation x^(3)_(1)+x^(3)_(2)+…+x^(3)_(n)+y^(3)_(1)+y^(3)_(2)+…+y^(3)_(m)=0,is studied.The number of solutions to this equation is A(n,m),for any variable x∈F_(q)^(*),y∈F q(i=1,2,…,n;j=1,2,…,m),the explicit expression of the equation generating function∑∞m=1∑∞n=1 A(n,m)x^(n)y^(m)is obtained.

关 键 词：有限域 GAUSS和 指数和 生成函数 

分 类 号：O156.2[理学—数学]