  The pentagram map is a natural iteration on projective equivalence classes of
(twisted) n-gons in the projective plane. It was recently proved ([OST]) that
the pentagram map is completely integrable, with the complete set of Poisson
commuting integrals given by the polynomials O1,...,O[n/2],On and
E1,...,E[n/2],En, previously constructed in [S3]. These polynomials are
somewhat reminiscent of the symmetric polynomials. It was observed in computer
experiments that if a polygon is inscribed into a conic then Oi=Ei for all i.
The goal of the paper is to prove this theorem. The proof is combinatorial, and
it was also suggested by computer experimentation.
