Poncelet's theorem for conics in any position and any characteristic

TL;DR

This paper proves Poncelet's theorem for any two conics in the projective plane over algebraically closed fields of any characteristic other than two, including cases where the conics are osculating.

Contribution

It extends Poncelet's theorem to all conic configurations over fields of any characteristic except two, including osculating conics.

Findings

Poncelet's theorem holds for any two conics in characteristic not two.

Infinitely many polygons exist inscribed and circumscribed in conics in these settings.

Special considerations are provided for characteristic two cases.

Abstract

Poncelet's theorem states that if there exists an n-sided polygon which is inscribed in a given conic C and circumscribed about another conic D, then there are infinitely many such n-gons. Proofs of this theorem that we are aware of, including Poncelet's original proof and the celebrated modern proof by Griffiths and Harris, assume the two conics to be in general position (that is, not tangent or at least not osculating), or be defined over the field of complex numbers, or both. Here we show that Poncelet's theorem holds for any two conics C and D in the projective plane over an algebraically closed field k of any characteristic other than two. If C and D are osculating and char(k)>2, our result shows that there always exist infinitely many polygons of char(k) sides that are inscribed in C and circumscribed about D. We also describe the situation in characteristic two in the appendix.