Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition

TL;DR

This paper refines the understanding of the density of conic pairs satisfying Poncelet conditions over finite fields, extending results from triangles to larger polygons and proposing a general conjecture involving divisor counts.

Contribution

It provides an exact formula for the density of conic pairs satisfying the Poncelet triangle condition and extends the analysis to n-gons, correcting previous conjectures and proposing a new general density conjecture.

Findings

Exact density formula for Poncelet triangle condition: (q-1)/(q^2 - q + 1)

Density for Poncelet tetragon condition: 1/q + O(q^{-3/2})

General density conjecture involving divisor counts d'(n)

Abstract

An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field $\mathbb{F}_q$ with characteristic greater than $3$, Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is $\frac{1}{q}+O(q^{-2})$. We improve this result, showing that the density is exactly $\frac{q-1}{q^2-q+1}$. We consider the problem of determining the density of pairs of conics satisfying the Poncelet $n$-gon condition for larger $n$. We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is $\frac{1}{q} + O(q^{-{3/2}})$. We show that when $n$ is an odd integer coprime to $q$, the density of pairs of smooth conics satisfying this condition is $\frac{d(n)-1}{q}+O(q^{-3/2})$, where $d(n)$ is the number of divisors of $n$. More generally, we conjecture that the density of pairs of conics satisfying the Poncelet $n$-gon condition is $d'(n)/q$ in general, where $d'(n)$ is the number of divisors of $n$ not equal to $1$ or $2$. Our argument involves analyzing the $n$-torsion points on a certain elliptic curve over the function field $K = \mathbb{F}_q(\lambda)$.