A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$

Year 2020, Volume: 13 Issue: 1, 21 - 40, 30.01.2020

Norbert Hungerbühler , Katharina Kusejko

https://doi.org/10.36890/iejg.590595

Abstract

We study Poncelet's Theorem in finite projective planes over the field GF(q), q = p^m for p an odd prime and m > 0, for a particular pencil of conics. We investigate whether we can find polygons with n sides which are inscribed in one conic and circumscribed around 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 Theorem 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(q) 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 plane PG(2,q). 

Keywords

Poncelet’s Theorem, finite projective planes, pencil of conics, quadratic residues

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