A Poncelet Criterion for Special Pairs of Conics in PG(2; pm)

Abstract

We study Poncelet’s Theorem in finite projective planes over the field GF(q), q = pm for p anodd prime and m 1, for a particular pencil of conics. We investigate whether we can findpolygons with n sides which are inscribed in one conic and circumscribed around the other, socalledPoncelet polygons. By using suitable elements of the dihedral group for these pairs, weprove that the length n of such Poncelet polygons is independent of the starting point. In thissense Poncelet’s Theorem is valid. By using Euler’s divisor sum formula for the totient functionwe determine the number of conic pairs which carry Poncelet polygons of length n. Moreover, weintroduce polynomials whose zeros in GF(q) yield information about the relation of a given pairof conics: In particular, we can decide for a given integer n, whether and how we can find Ponceletn-gons for pairs of conics in the projective plane PG(2; q).