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Journals / International Electronic Journal of Geometry / 2026 / Volume: 13 Issue: 1

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

Norbert Hungerbühler(ETH Zürich)0000-0001-6191-0022
Katharina Kusejko(Universitätsspital Zürich)0000-0002-4638-1940
Pages
—
Publication Date
2020-01-30
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

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 ).

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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.
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Key Findings & Data:Moreover, we will introduce polynomials whose zeros in GF ( q ) yield information about the relation of a given pair of conics.
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Conclusion & Impact: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 ).
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Poncelet’s Theorem · finite projective planes · pencil of conics

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 ).

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Cite This Article

Hungerbühler, N., Kusejko, K. (2020). A Poncelet Criterion for Special Pairs of Conics in $PG(2,p^m)$. International Electronic Journal of Geometry, Volume: 13 Issue: 1. https://doi.org/10.36890/iejg.590595

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