On the Poncelet triangle condition over finite fields

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.