有限域上的高斯和与有理点的计算

Gauss sums and computation of rational points in finite fields

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.