摘要
设f是有限域Fq上的n元m项多项式,D_f∈Z_(≥0)^(n×m)为其次数矩阵,用N(f)表示由超曲面f=0在仿射空间An(Fq)确定的Fq-有理点的个数.若矩阵A∈Z^(n×m)在环Z/(q-1)Z中与Df行等价,则记为D_fr~A.本文利用高斯和给出了当m≤n且0<D_fr~diag(λ_1,···,λ_m),其中λi∈{1,3}时N(f)的具体表达式,从而推广了已知结论.
Let f be a polynomial over the finite field Fq in n variables with degree matrix Df∈Z≥0^n×m and N(f) be the number of Fq-rational points on the hypersurface defined by f = 0 in the affine space An(Fq). For an A∈Z^n×m, let D.f - A denote that D f is row equivalent to A in the ring Z/(q - 1)Z. In this paper, by using Gauss sums we obtain the formula for N(f) when m 〈 n and 0 〈 D f - diag(λ1,... , λm) with λi ∈ {1, 3}, which generalizes the perviously known results.
出处
《纯粹数学与应用数学》
2017年第6期634-643,共10页
Pure and Applied Mathematics
基金
国家自然科学基金(11371208)
宁波市自然科学基金(2017A610134)
关键词
有限域
有理点
特征和
高斯和
finite field, rational point, character sum, Gauss sum