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

Year 2020, , 21 - 40, 30.01.2020

Norbert Hungerbühler , Katharina Kusejko

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

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

References

• [1] Abatangelo, V., Fisher, J. C., Korchmáros, G., Larato, B.: On the mutual position of two irreducible conics in PG(2; q), q odd. Adv. Geom. 11 (4), 603–614 (2011).
• [2] Berger, M.: Geometry II, Universitext. Springer-Verlag, Berlin (1987).
• [3] Bos, H. J. M., Kers, C., Oort, F., Raven, D.W.: Poncelet’s closure theorem. Exposition. Math. 5 (4), 289–364 (1987).
• [4] Cayley, A.: Developments on the porism of the in-and-circumscribed polygon. Philosophical magazine. 7 (4), 289–364 (1854).
• [5] Dragović, V., Radnović, M.: Poncelet porisms and beyond. Frontiers in Mathematics. Birkhäuser/Springer Basel AG, Basel (2011).
• [6] Griffiths, P., Harris, J.: On Cayley’s explicit solution to Poncelet’s porism. Enseign. Math. 24 (1–2), 31–40 (1978).
• [7] Halbeisen, L., Hungerbühler, N.: A Simple Proof of Poncelet’s Theorem. Amer. Math. Monthly. 122 (6), 603–614 (2015).
• [8] Hardy, G. H., Wright, E. M.: An introduction to the theory of numbers. Oxford University Press, Oxford (2008).
• [9] Hirschfeld, J.W. P.: Projective geometries over finite fields. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1998).
• [10] Hungerbühler, N., Kusejko, K.: Poncelet’s Theorem in the four non-isomorphic finite projective planes of order 9. Ars Combin. 140, 21–44 (2018).
• [11] Korchmáros, G., Sz˝onyi, T.: Affinely regular polygons in an affine plane. Contrib. Discrete Math. 3 (1), 20–38 (2008).
• [12] Kusejko, K.: Simultaneous diagonalization of conics in PG(2; q). Des. Codes Cryprogr. 79 (3), 565–581 (2016).
• [13] Luisi, G.: On a theorem of Poncelet. Atti Sem. Mat. Fis. Univ. Modena. 31 (2), 341–347 (1984).
• [14] Poncelet, J.-V.: Traité des propriétés projectives des figures. Tome II. Les Grands Classiques Gauthier-Villars. Reprint of the second (1866) edition. Éditions Jacques Gabay, Sceaux (1995).