A Poncelet Criterion for special pairs of conics in $PG(2,p)$

TL;DR

This paper investigates Poncelet's Theorem for pairs of conics in finite projective planes over GF(p), establishing conditions for the existence of Poncelet Polygons and analyzing their properties using group theory and quadratic residue theory.

Contribution

It introduces a criterion for Poncelet Polygons in finite projective planes, linking their existence to polynomial zeros and quadratic residues, and extends classical results to finite fields.

Findings

Poncelet's Porism holds for certain conic pairs in GF(p).

The number of conic pairs with Poncelet Polygons of length n is characterized.

Polynomial conditions determine the existence of Poncelet Polygons for given conic pairs.

Abstract

We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are inscribed in one conic and circumscribed about the other, so-called Poncelet Polygons. By using suitable elements of the dihedral group for these pairs, we prove that the length $n$ of such Poncelet Polygons is independent of the starting point. In this sense Poncelet's Porism is valid. By using Euler's divisor sum formula for the totient function, we can make a statement about the number of different conic pairs, which carry Poncelet Polygons of length $n$. Moreover, we will introduce polynomials whose zeros in $GF(p)$ yield information about the relation of a given pair of conics. In particular, we can decide for a given integer $n$, whether and how we can find Poncelet Polygons for pairs of conics in the given coordinate plane. We will see that this condition is closely connected with the theory of quadratic residues.