高斯和与有理点的计算  

Gauss sums and computation of rational points

作　　者：姜坤 高伟 曹炜 JIANG Kun;GAO Wei;CAO Wei(School of Mathematics and Statistics,Ningbo University,Ningbo 315211,China)

机构地区：[1]宁波大学数学与统计学院

出　　处：《宁波大学学报（理工版）》2020年第1期65-68,共4页Journal of Ningbo University：Natural Science and Engineering Edition

基　　金：国家自然科学基金(11871291);宁波市自然科学基金(2019A610035)

摘　　要：计算有限域上代数簇有理点个数是有限域研究中的重要课题.设Fq为q元有限域,f是Fq上的非零多项式, Df为其次数矩阵,用N(f)表示超曲面f=0在Fq上的有理点个数.若Df在剩余类环Z/(q-1)Z中与整数矩阵A行等价,则记为DfqA.利用高斯和给出了当Dfq diag(λ1,···,λm),其中λi∈{1, p1},p1为q-1的一个素因子时N(f)的具体表达式,从而推广了已知的结论.It is important in the area of finite fields study to compute the number of rational points on algebraic varieties in finite fields. Let Fq be the finite field of order q and f be a nonzero polynomial over Fq with degree matrix Df;denote by N( f) the number of Fq-rational points of the hypersurface defined by f =0;write Dfq A when Df is row equivalent to an integer matrix A in the residue ring Z/(q -1)Z. Using Gauss sums, we obtain the explicit formula for N( f) in the case of Dfq diag(λ1,···,λ--m), where λi∈{1, p1 } and p1 is a prime divisor of q-1, which generalizes the known results.

关 键 词：有限域 有理点 高斯和 特征 

分 类 号：O156.1[理学—数学]