# The pentagram map, Poncelet polygons, and commuting difference operators

## TL;DR

This paper characterizes Poncelet polygons as convex polygons that are projectively equivalent to their pentagram map images, using commuting difference operators and elliptic curve theory.

## Contribution

It provides a new characterization of Poncelet polygons in the convex case based on projective equivalence with their pentagram images, employing advanced algebraic and geometric tools.

## Key findings

- Convex polygons projectively equivalent to their pentagram images are Poncelet polygons.
- The proof utilizes commuting difference operators and properties of elliptic curves.
- The characterization extends previous results relating pentagram maps and Poncelet polygons.

## Abstract

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, i.e. polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of R. Schwartz says that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$. In the present paper we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.