The Pentagram Integrals on Inscribed Polygons

The pentagram map is a completely integrable system defined on the moduli space of polygons. The integrals for the system are certain weighted homogeneous polynomials, which come in pairs: $E_1,O_2,E_2,O_2,\dots$ In this paper we prove that $E_k=O_k$ for all $k$, when these integrals are restricted to the space of polygons which are inscribed in a conic section. Our proof is essentially a combinatorial analysis of the integrals.