On Probabilities regarding Poncelet Polygons over Finite Fields

Abstract

An n-sided polygon that is inscribed in a conic A and circumscribed about a conic B is called a Poncelet polygon, and we call the pair of conics (A,B) an n-Poncelet pair. In the projective plane over a finite field of characteristic not equal to 2, we study Poncelet polygons and n-Poncelet pairs, with emphasis on the cases n = 3 and n=4. In particular, we discuss the construction of Poncelet polygons and derive results regarding degenerate Poncelet polygons. Moreover, we provide in-depth results regarding the construction of Poncelet triangles. For our main result, we compute the probability of obtaining a 3-Poncelet pair or a 4-Poncelet pair when we randomly select a pair of distinct conics (A,B), with A smooth or singular and B smooth, in a fixed pencil of conics. We do this for all pencils, classified up to projective automorphism, with at least one smooth conic.