The Algebraic Dynamics of the Pentagram Map

TL;DR

This paper extends the understanding of the pentagram map, a discrete dynamical system on polygons, by proving its algebraic complete integrability over arbitrary algebraically closed fields, generalizing previous complex results.

Contribution

It proves the algebraic complete integrability of the pentagram map over any algebraically closed field of characteristic not 2, generalizing prior complex field results.

Findings

Pentagram map is a discrete integrable system on twisted polygons.

Constructs the moduli space of twisted polygons.

Derives formulas and Lax representation independent of characteristic.

Abstract

The pentagram map, introduced by Schwartz in 1992, is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary algebraically closed field of characteristic not equal to 2. We prove that the pentagram map on twisted polygons is a discrete integrable system, in the sense of algebraic complete integrability: the pentagram map is birational to a self-map of a family of abelian varieties. This generalizes Soloviev's proof of complex integrability. In the course of the proof, we construct the moduli space of twisted $n$-gons, derive formulas for the pentagram map, and calculate the Lax representation by characteristic-independent methods.