- International Electronic Journal of Geometry
- Vol: 13 Issue: 1
- A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$
A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$
Authors : Norbert Hungerbühler, Katharina Kusejko
Pages : 21-40
Doi:10.36890/iejg.590595
View : 6 | Download : 3
Publication Date : 2020-01-30
Article Type : Research
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