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

#### Norbert HUNGERBÜHLER [1] , Katharina KUSEJKO [2]

We study Poncelet's Theorem in finite projective planes over the field GF(q), pm 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).

Poncelet’s Theorem, finite projective planes, pencil of conics, quadratic residues
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Primary Language en Mathematics Research Article Orcid: 0000-0001-6191-0022Author: Norbert HUNGERBÜHLER (Primary Author)Institution: ETH ZürichCountry: Switzerland Orcid: 0000-0002-4638-1940Author: Katharina KUSEJKO Institution: Universitätsspital ZürichCountry: Switzerland Publication Date : January 30, 2020
 Bibtex @research article { iejg590595, journal = {International Electronic Journal of Geometry}, issn = {}, eissn = {1307-5624}, address = {}, publisher = {Kazım İLARSLAN}, year = {2020}, volume = {13}, pages = {21 - 40}, doi = {10.36890/iejg.590595}, title = {A Poncelet Criterion for Special Pairs of Conics in \$PG(2,p\^m)\$}, key = {cite}, author = {HUNGERBÜHLER, Norbert and KUSEJKO, Katharina} } APA HUNGERBÜHLER, N , KUSEJKO, K . (2020). A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$. International Electronic Journal of Geometry , 13 (1) , 21-40 . DOI: 10.36890/iejg.590595 MLA HUNGERBÜHLER, N , KUSEJKO, K . "A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$". International Electronic Journal of Geometry 13 (2020 ): 21-40 Chicago HUNGERBÜHLER, N , KUSEJKO, K . "A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$". International Electronic Journal of Geometry 13 (2020 ): 21-40 RIS TY - JOUR T1 - A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$ AU - Norbert HUNGERBÜHLER , Katharina KUSEJKO Y1 - 2020 PY - 2020 N1 - doi: 10.36890/iejg.590595 DO - 10.36890/iejg.590595 T2 - International Electronic Journal of Geometry JF - Journal JO - JOR SP - 21 EP - 40 VL - 13 IS - 1 SN - -1307-5624 M3 - doi: 10.36890/iejg.590595 UR - https://doi.org/10.36890/iejg.590595 Y2 - 2020 ER - EndNote %0 International Electronic Journal of Geometry A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$ %A Norbert HUNGERBÜHLER , Katharina KUSEJKO %T A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$ %D 2020 %J International Electronic Journal of Geometry %P -1307-5624 %V 13 %N 1 %R doi: 10.36890/iejg.590595 %U 10.36890/iejg.590595 ISNAD HUNGERBÜHLER, Norbert , KUSEJKO, Katharina . "A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$". International Electronic Journal of Geometry 13 / 1 (January 2020): 21-40 . https://doi.org/10.36890/iejg.590595 AMA HUNGERBÜHLER N , KUSEJKO K . A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$. Int. Electron. J. Geom.. 2020; 13(1): 21-40. Vancouver HUNGERBÜHLER N , KUSEJKO K . A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$. International Electronic Journal of Geometry. 2020; 13(1): 40-21.

Authors of the Article
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