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

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Pith's one-line read For two transversal smooth conics over a finite field, the exact numbers of points in the four internal/external position classes are explicit rational expressions in Frobenius traces of two associated elliptic curves and two intersection counts.

arxiv 2608.22961 v1 pith:75XKJA2Q submitted 2026-08-24 math.CO math.NT

Tianhao Wang This is my paper · ORCID
classification math.COmath.NT
keywords mathcalmathbbconicsexactasgarliexternalformulainternal
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The reading

Finite fields are number systems with finitely many elements. In the projective plane over such a field, a conic is a curve given by a quadratic equation, so it behaves like an oval. A point away from a conic is called external when the two tangent lines from that point to the conic are both defined over the field, and internal when those tangents only become visible after passing to a quadratic extension. Given two conics that meet transversely, the paper counts the points in the plane that are internal or external to each conic. Before this paper, only approximate counts with an error of size about q to the power three-halves were known.

The proof slices the plane using all tangent lines to the first conic. On a tangent line, whether a point is internal to the second conic depends only on whether that line meets the second conic in two field-rational points or two conjugate points. The collection of all such line-by-line data is packaged into an incidence variety, and this variety turns out to have genus one. The key step identifies those genus-one curves, over the ground field, with two explicit elliptic curves obtained from the determinant of the pencil formed by the two conic matrices. Once the identification is made, the four exact counts are rational expressions in the Frobenius traces of these elliptic curves and in the numbers of rational intersection points of the conics and of their duals.

For one of the four classes, points external to the first conic and internal to the second, the formula is one quarter of q squared minus b2 q minus 1 minus t1 plus b1 plus t2, where t1 and t2 are elliptic curve traces and b1 and b2 are intersection counts. The formula implies that every class is q squared over four plus an error of size at most q plus sqrt(q) plus 1.

Extended reading notes

Core claim

Theorem 2.2: for two smooth transversal conics over F_q, N_ext-int(C,D) = (1/4)(q^2 - b2 q - 1 - t1 + b1 + t2), with analogous exact formulas for the other three internal/external classes, and each count differs from q^2/4 by at most q + sqrt(q) + 1. If correct, prior asymptotic estimates are replaced by exact arithmetic data: the Frobenius traces t1,t2 of two explicitly defined elliptic curves and the rational intersection counts b1,b2 of the conics and their duals.

Load-bearing premise

Lemma 2.3 asserts that the incidence variety E1 = {(P,l) in C x D* : P in l} is isomorphic over F_q to the elliptic curve y^2 = det(D) det(xC + D), not merely over the algebraic closure. This is the load-bearing twist identification: if the quadratic twist were wrong, t1 and t2 would be traces of different elliptic curves and all four formulas in Theorem 2.2 would fail. The proof depends on the Cremona-Fisher-Stoll Jacobian formulas for a binary quartic and on a direct determinant/change-of-variables computation, making it the most fragile step in the chain.

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Formalized claims in Lean

  1. Claim #1: Theorem 2.2: for two smooth transversal conics over F_q, N_ext-int(C,D) = (1/4)(q^2 - b2 q - 1 - t1 + b1 + t2), with analogous exact formulas for the other three internal/external classes, and each count differs from q^2/4 by at most q + sqrt(q) + 1. If correct, prior asymptotic estimates are replaced by exact arithmetic data: the Frobenius traces t1,t2 of two explicitly defined elliptic curves an

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Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim has no fitted or hand-chosen constants. The auxiliary elliptic curves E1 and E2 are fully determined by the conic matrices C and D, while b1, b2, t1, and t2 are structural data of the input pair. All listed axioms are standard background results or cited lemmas, invoked in the incidence-variety and elliptic-curve steps.

assumptions (6)
  • standard math A point not on a smooth conic has either two F_q-rational tangents or two Galois-conjugate tangents, giving the external/internal dichotomy (Definition 1.1).
    Used throughout as the basic classification of points; follows from conic geometry over finite fields.
  • standard math Transversality of two conics is equivalent to det(xC+zD) having three distinct roots in P^1 (Harris, Prop. 22.34).
    Invoked in Lemma 2.1 to transfer transversality to the dual conics.
  • standard math The discriminant governing rationality of the two tangents from P to D is -det(D) Q_D(P) up to a square (Asgarli-Yip, Lemma 2.4).
    Basis for the binary quartic model W^2 = -delta f(u,v) in Lemma 2.3.
  • standard math The Cremona-Fisher-Stoll formulas give the Jacobian of a binary quartic and are valid over fields of odd characteristic.
    Used to compute the Weierstrass equation (2) and match it to the determinantal cubic.
  • standard math Every smooth projective curve of genus 0 over F_q is isomorphic to P^1 and has q+1 rational points, and every genus-one curve over F_q has a rational point.
    Used in the double-cover incidence count on a line and to identify E1 and E2 with their Jacobians.
  • standard math Hasse bound |t_i| <= 2 sqrt(q) for elliptic curve traces.
    Used in the final error-term estimate to bound the difference from q^2/4.

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Pith. "Pith review of An Exact Counting Formula for the Mutual Position of Two Plane Conics." pith.science (2026). https://pith.science/paper/75XKJA2Q

@misc{pith2026260822961,
  author       = {Pith},
  title        = {Pith review of: An Exact Counting Formula for the Mutual Position of Two Plane Conics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75XKJA2Q}},
  note         = {Machine review of arXiv:2608.22961}
}
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.

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