- International Electronic Journal of Geometry
- Volume: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 : 46 | Download : 13
Publication Date : 2020-01-30
Article Type : Research Paper
Abstract :We study Poncelet`s Theorem in finite projective planes over the field GF insert ignore into journalissuearticles values( 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 insert ignore into journalissuearticles values( 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 insert ignore into journalissuearticles values(2, q );.Keywords : Poncelets Theorem, finite projective planes, pencil of conics, quadratic residues
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