有限域上对角三次方程解的个数

Number of Solutions of Diagonal Cubic Equations over Finite Fields

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.