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

Yıl 2020, Cilt: 13 Sayı: 1, 21 - 40, 30.01.2020

Norbert Hungerbühler Katharina Kusejko

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

Öz

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). 

Anahtar Kelimeler

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

Kaynakça

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