On the Poncelet triangle condition over finite fields

TL;DR

This paper investigates the probability that a random pair of conics over a finite field satisfies the Poncelet triangle condition, showing it approaches 1/q, and extends the discussion to polygons up to nine sides.

Contribution

It establishes the asymptotic probability for the Poncelet triangle condition over finite fields and conjectures similar probabilities for polygons with more sides based on computational evidence.

Findings

Probability of satisfying the triangle condition approaches 1/q

Established asymptotic behavior for triangles over finite fields

Conjectured probabilities for polygons with up to nine sides

Abstract

Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in ${\mathbf P}^2({\overline{\mathbb F}}_q)$, they intersect transverally and there exists a triangle inscribed in ${\mathcal A}$ and circumscribed around ${\mathcal B}$. It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability $1/q$. We also make a conjecture based upon computer experimentation which predicts this probability for tetragons, pentagons and so on up to enneagons.