论文标题
关于Artin-Schreier曲线和Hypersurfaces的理性点数
On the number of rational points of Artin-Schreier curves and hypersurfaces
论文作者
论文摘要
令$ \ mathbb f_ {q^n} $用$ q^n $元素表示有限字段。在本文中，我们确定$ \ Mathbb f_ {q^n} $ - 由$ y^q-y = x（x^^{q^i} -X）-λ$和Artin-Schreier HyperSurface $ y HyperSurface $ y^Q-y y^q-y y = sum_ = \ sum_ = \ sum_ = \ sum_ = \ sum_ = \ sum_ = \ sum_ = 1} a_jx_j（x_j^{q^{i_j}} - x_j）-λ。$此外，在两种情况下，我们都表明，在$ \ mathbb f_q $以上的$λ\ in \ mathbb f_ {q^n} $的$λ\ in \ mathbb f_ {q^n}的跟踪中仅获得了weil键。我们使用二次形式和置换矩阵来确定这些曲线和超曲面的仿射合理点的数量。
Let $\mathbb F_{q^n}$ denote the finite field with $q^n$ elements. In this paper we determine the number of $\mathbb F_{q^n}$-rational points of the affine Artin-Schreier curve given by $y^q-y = x(x^{q^i}-x)-λ$ and of the Artin-Schreier hypersurface $y^q-y=\sum_{j=1}^r a_jx_j(x_j^{q^{i_j}}-x_j)-λ.$ Moreover in both cases, we show that the Weil bound is attained only in the case where the trace of $λ\in\mathbb F_{q^n}$ over $\mathbb F_q$ is zero. We use quadratic forms and permutation matrices to determine the number of affine rational points of these curves and hypersurfaces.
