论文标题
$ \ mathbb f_q $ - 线性多项式给出的Artin-Schreier曲线
Artin-Schreier curves given by $\mathbb F_q$-linearized polynomials
论文作者
论文摘要
令$ \ mathbb f_q $为带有$ q $元素的有限字段，其中$ q $是奇数prime $ p $的力量。在本文中，我们将循环矩阵和二次形式与artin -schreier曲线相关联$ y^q -y = x \ cdot f（x） - λ，$，其中$ f（x）$是$ \ mathbb f_q $ learearized tolearized多项式和$λ\ in \ mathbb f_q $。我们的结果提供了该曲线的仿射合理点数的表征，$ \ mathbb f_q $，$ \ gcd（q，r）= 1 $。在情况下，在$ f（x）= x^{q^i} -x $中，我们完整地描述了Legendre符号和二次字符的仿射理性点数。
Let $\mathbb F_q$ be a finite field with $q$ elements, where $q$ is a power of an odd prime $p$. In this paper we associate circulant matrices and quadratic forms with the Artin-Schreier curve $y^q - y= x \cdot F(x) - λ,$ where $F(x)$ is a $\mathbb F_q$-linearized polynomial and $λ\in \mathbb F_q$. Our results provide a characterization of the number of affine rational points of this curve in the extension $\mathbb F_{q^r}$ of $\mathbb F_q$, for $\gcd(q,r)=1$. In the case $F(x) = x^{q^i}-x$ we give a complete description of the number of affine rational points in terms of Legendre symbols and quadratic characters.
