Poncelet Triangles and Tetragons over Finite Fields

TL;DR

This paper calculates the probability that a randomly chosen pair of conics over a finite field admits inscribed Poncelet triangles or tetragons, considering various configurations and classifications of conic pencils.

Contribution

It provides a comprehensive probability analysis for Poncelet polygons over finite fields for all conic pencils with at least one smooth conic, including non-transversal intersections.

Findings

Probability formulas for inscribed Poncelet triangles and tetragons.

Classification of conic pencils with smooth and singular conics.

Analysis includes non-transversal intersection cases.

Abstract

In the projective plane over a finite field of characteristic not equal to 2, we compute the probability that a randomly selected pair of distinct conics $(\mathscr{A},\mathscr{B})$, with $\mathscr{A}$ smooth or singular and $\mathscr{B}$ smooth, in a fixed pencil of conics will admit a triangle or a tetragon inscribed in $\mathscr{A}$ and circumscribed about $\mathscr{B}$. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic; effectively allowing the case where our conic pairs intersect non-transversally.