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Pentagram Rigidity for Centrally Symmetric Octagons

Richard Evan Schwartz

Abstract

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

Pentagram Rigidity for Centrally Symmetric Octagons

Abstract

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.
Paper Structure (34 sections, 35 theorems, 93 equations)

This paper contains 34 sections, 35 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. The Main Ideas
  4. Complete Integrability
  5. The Map in Coordinates
  6. Some Important Functions
  7. Proof of Theorem \ref{['integrable']}
  8. Positivity
  9. Duality and Convexity
  10. Integral Curves
  11. Proof of the Main Theorem: Outline
  12. Linear Independence
  13. The Dependence Set
  14. Resultants
  15. The Proof modulo the Non-Vanishing Lemma
  16. ...and 19 more sections

Key Result

Theorem 1.2

The Pentagram Rigidity Conjecture is true for $(8,3)$ provided that the octagon is centrally symmetric.

Theorems & Definitions (36)

  • Conjecture 1.1: Pentagram Rigidity
  • Theorem 1.2: Main
  • Theorem 1.3
  • Theorem 1.4: Integrability
  • Lemma 2.1: Positivity
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 3.1
  • Lemma 3.2: Non-Vanishing
  • ...and 26 more