The Pentagram Integrals on Inscribed Polygons
Combinatorics
2010-04-27 v1 Algebraic Geometry
Abstract
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.
Keywords
Cite
@article{arxiv.1004.4311,
title = {The Pentagram Integrals on Inscribed Polygons},
author = {Richard Evan Schwartz and Serge Tabachnikov},
journal= {arXiv preprint arXiv:1004.4311},
year = {2010}
}