论文标题
一类超电子曲线的理性点数
The number of rational points of a class of superelliptic curves
论文作者
论文摘要
在本文中，我们研究了Aggine Curve $ \ Mathcal $ \ MathCal {x} _ {d，a，a，b} $的$ \ MATHBB f_ {q^n} $ - 由等式$$ y^y^d = y^d = ax \ ax \ text {tr}（tr}（tr}（x）+b，$ \ wery $ \ tre tr trace trace trace， f_ {q^n} $ to $ \ mathbb f_ {q} $和$ d $是一个正整数。特别是，我们提出了$ \ mathbb f_ {q} $的数量的界限 - $ \ mathcal {x} _ {d，a，b} $以及$ d $满足自然条件的情况，获得了理性点数量的明显格式。特别是，给出了$ d = 2 $的完整表征。由于我们的结果，我们计算$ \ Mathbb f_ {q^n} $中的元素$α$的数量，以便$ \ Mathbb f_ {q^n} $中的$α$和$α$和$α$和$α$和$ \ text {tr}（α）$。
In this paper, we study the number of $\mathbb F_{q^n}$-rational points on the affine curve $\mathcal{X}_{d,a,b}$ given by the equation $$ y^d=ax\text{Tr}(x)+b,$$ where $\text{Tr}$ denote the trace function from $\mathbb F_{q^n}$ to $\mathbb F_{q}$ and $d$ is a positive integer. In particular, we present bounds for the number of $\mathbb F_{q}$-rational points on $\mathcal{X}_{d,a,b}$ and, for the cases where $d$ satisfies a natural condition, explicit formulas for the number of rational points are obtained. Particularly, a complete characterization is given for the case $d=2$. As a consequence of our results, we compute the number of elements $α$ in $\mathbb F_{q^n}$ such that $α$ and $\text{Tr}(α)$ are quadratic residues in $\mathbb F_{q^n}$.
