An Exact Counting Formula for the Mutual Position of Two Plane Conics

Abstract

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.