The Pentagram Integrals on Inscribed Polygons

TL;DR

This paper proves that for polygons inscribed in a conic, certain integrals of the pentagram map are equal, confirming a conjecture suggested by computational experiments and using combinatorial methods.

Contribution

It provides a rigorous combinatorial proof that inscribed polygons in a conic satisfy Oi=Ei for all i, confirming a conjecture from computer experiments.

Findings

Polygons inscribed in a conic satisfy Oi=Ei for all i.

The proof is combinatorial and confirms a conjecture from computational observations.

Supports the integrability properties of the pentagram map for inscribed polygons.

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.