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

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

Problem Statement
The paper addresses a combinatorial geometry problem over finite fields: determining the exact number of $\mathbb{F}_q$-rational points in the projective plane $\mathbb{P}^2(\mathbb{F}_q)$ that occupy specific mutual positions relative to two smooth plane conics, $\mathcal{C}$ and $\mathcal{D}$. Specifically, the goal is to count points that are simultaneously internal or external to $\mathcal{C}$ and internal or external to $\mathcal{D}$.

The definitions of "internal" and "external" rely on the rationality of tangent lines: a point $P \notin \mathcal{C}$ is external if the two tangent lines from $P$ to $\mathcal{C}$ are defined over $\mathbb{F}_q$, and internal if they are Galois-conjugate lines defined over $\mathbb{F}_{q^2}$ but not $\mathbb{F}_q$.

Previous work by Asgarli and Yip established asymptotic estimates for these counts, showing that the number of points in any specific configuration (e.g., external to $\mathcal{C}$ and internal to $\mathcal{D}$) is approximately $q^2/4$ with an error term of order $O(q^{3/2})$. Slavov later provided a geometric reinterpretation using multiplicative character sums and the Lang–Weil bound, improving the explicit constants but retaining the $O(q^{3/2})$ error magnitude.

Methodology
The author employs a geometric approach centered on the analysis of incidence varieties associated with the pair of conics and their duals. The core strategy involves:

1. Incidence Varieties and Elliptic Curves: The paper constructs two incidence varieties, $E_1$ and $E_2$, defined as:

   • $E_1 = \{(P, L) \in \mathcal{C} \times \mathcal{D}^* : P \in L\}$
   • $E_2 = \{(L, P) \in \mathcal{C}^* \times \mathcal{D} : P \in L\}$
     where $\mathcal{C}^*$ and $\mathcal{D}^*$ are the dual conics. The projection maps from these varieties to the conics (or their duals) are shown to be 2-to-1 covers branched at the intersection points. Under the assumption of transversal intersection, these varieties are proven to be elliptic curves.

2. Explicit Models: The paper derives explicit Weierstrass models for these elliptic curves over $\mathbb{F}_q$. If $C$ and $D$ are the symmetric matrices representing $\mathcal{C}$ and $\mathcal{D}$, the curves are identified as:

   • $E_1: y^2 = \det(D) \det(xC + D)$
   • $E_2: y^2 = \det(C) \det(xD + C)$
     The author provides a detailed proof (Lemma 2.3) establishing the isomorphism between the incidence variety $E_1$ and the curve defined by the determinant polynomial, resolving the specific quadratic twist required over non-algebraically closed fields.

3. Counting via Frobenius Traces: The counting problem is reduced to analyzing the rational points on these elliptic curves. The number of points on a tangent line to $\mathcal{C}$ that are internal/external to $\mathcal{D}$ depends on whether the intersection points of the line with $\mathcal{D}$ are rational or conjugate. This distribution is governed by the Frobenius traces ($t_1, t_2$) of $E_1$ and $E_2$, and the number of rational intersection points between the conics ($b_1$) and their duals ($b_2$).

4. Summation and Symmetry: By summing over all tangent lines and utilizing the symmetry between the primal and dual configurations, the author derives exact linear combinations of $q$, $b_1, b_2, t_1,$ and $t_2$ to determine the counts for all four mutual position configurations.

Key Contributions and Results
The primary contribution is the derivation of exact formulas for the number of points in each configuration, replacing previous asymptotic estimates. Let $N_{\text{ext-int}}(\mathcal{C}, \mathcal{D})$ denote the number of points external to $\mathcal{C}$ and internal to $\mathcal{D}$, and similarly for other combinations. The main theorem states:

$$\begin{aligned} N_{\text{ext-int}} &= \frac{1}{4}(q^2 - b_2 q - 1 - t_1 + b_1 + t_2) \\ N_{\text{int-ext}} &= \frac{1}{4}(q^2 - b_2 q - 1 + t_1 + b_1 - t_2) \\ N_{\text{ext-ext}} &= \frac{1}{4}(q^2 + b_2 q - 1 + t_1 + b_1 + t_2) \\ N_{\text{int-int}} &= \frac{1}{4}(q^2 + b_2 q - 1 - t_1 + b_1 - t_2 - 4q) \end{aligned}$$

Additionally, the paper provides exact formulas for points on $\mathcal{C}$ that are internal/external to $\mathcal{D}$, refining bounds from previous literature.

Significance and Error Analysis
The paper highlights a significant reduction in the error term compared to higher-dimensional or general multi-quadratic cover approaches.

• Error Reduction: While general methods (e.g., Slavov's multi-quadratic cover framework) yield an error term of $O(q^{3/2})$ due to the dimension of the base space ($\mathbb{P}^2$), this planar conic setting achieves an error term bounded by $|N - q^2/4| \leq q + \sqrt{q} + 1$.
• Reason for Improvement: The author attributes this improvement to the specific geometry of plane conics: every $\mathbb{F}_q$-rational point on an $\mathbb{F}_q$-tangent line to $\mathcal{C}$ is automatically external to $\mathcal{C}$. This allows the problem to be sliced into a one-dimensional family of lines, reducing the cohomological error from $O(q^{3/2})$ to $O(q^{1/2})$ (via the Hasse bound on the elliptic curves) plus a linear term $O(q)$ arising from the intersection counts.

Limitations and Generalization
The paper explicitly notes that this method is specific to the planar case. The author argues that generalizing this exact counting formula to higher-dimensional quadrics is difficult because a rational point on a rational tangent space of a higher-dimensional quadric is not automatically external to the quadric. Consequently, the reduction to a one-dimensional family does not hold, and the error terms would likely revert to those governed by higher étale cohomology groups, preventing a similar exact formula.

The author provides a Sage worksheet to verify the formulas and acknowledges the use of AI tools for grammar and symbolic computation organization, while taking full responsibility for the mathematical content.