平面二次曲线相互位置的精确计数公式
An Exact Counting Formula for the Mutual Position of Two Plane Conics
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中文总结 AI 辅助
该研究针对有限域上两条横截相交的光滑平面二次曲线，给出了同时在两条曲线内部/外部的射影有理点数量的精确公式，改进了已有渐近估计并关联椭圆曲线的Frobenius迹，还讨论了推广至高维二次曲面的挑战。
中文摘要 AI 辅助
设q为奇素数幂，$\boldsymbol{\textit{C}}$、$\boldsymbol{\textit{D}}$是定义在有限域$\boldsymbol{\textit{F}}_q$上的两条横截相交的光滑平面二次曲线。我们给出了射影平面$\boldsymbol{\textit{P}}^2(\boldsymbol{\textit{F}}_q)$中同时在$\boldsymbol{\textit{C}}$内部/外部且在$\boldsymbol{\textit{D}}$内部/外部的点的数量的精确公式，这改进了Asgarli和Yip[文献1.2]中对该量的渐近估计$\frac{q^2}{4}+O(q^{3/2})$，特别地，我们证明误差项的大小至多为$q+\boldsymbol{\textit{q}}^{1/2}+1$。通过研究与该问题相关的关联簇的几何，我们将精确点计数直接与两条相关椭圆曲线的Frobenius迹、$\boldsymbol{\textit{C}}$与$\boldsymbol{\textit{D}}$的$\boldsymbol{\textit{F}}_q$有理交点数量，以及对应的对偶二次曲线$\boldsymbol{\textit{C}}^*$与$\boldsymbol{\textit{D}}^*$的交点数量联系起来。最后，我们给出一个注记，解释将该方法推广到高维二次曲面研究时面临的挑战。
英文摘要
Let $q$ be an odd prime power, and $\mathcal{C}, \mathcal{D}$ be two smooth plane conics defined over $\mathbb{F}_q$ with transversal intersection. We present an exact formula for the number of points in $\mathbb{P}^2(\mathbb{F}_q)$ that are internal/external to $\mathcal{C}$ and internal/external to $\mathcal{D}$. This refines the $\frac{q^2}{4}+O(q^{3/2})$ asymptotic estimate for this quantity due to Asgarli and Yip \cite[Theorem 1.2]{Asgarli}. In particular, we show that the error term is of size at most $q+\sqrt{q} + 1$. By studying the geometry of the incidence variety related to this problem, we link the exact point counts directly to the Frobenius traces of two associated elliptic curves, and the number of $\mathbb{F}_q$-rational intersection points of $\mathcal{C}$ and $\mathcal{D}$ and of the corresponding dual conics $\mathcal{C}^*$ and $\mathcal{D}^*$. Lastly, We provide a remark explaining the challenges in generalizing this method to the study of higher-dimensional quadrics.
