Ehrhart polynomials and symplectic embeddings of ellipsoids

Abstract

McDuff and Schlenk determined when a four-dimensional ellipsoid can be symplectically embedded into a ball, and found that part of the answer is given by an infinite "Fibonacci staircase." Similarly, Frenkel and Müller determined when a four-dimensional ellipsoid can be symplectically embedded into the ellipsoid E(1,2) and found that part of the answer is given by a "Pell staircase." ECH capacities give an obstruction to symplectically embedding one four-dimensional ellipsoid into another, and McDuff showed that this obstruction is sharp. We use this result to give new proofs of the staircases of McDuff-Schlenk and Frenkel-Müller, and we prove that another infinite staircase arises for embeddings into the ellipsoid E(1,3/2). Our proofs relate these staircases to a combinatorial phenomenon of independent interest called "period collapse" of the Ehrhart quasipolynomial. In the appendix, we use McDuff's theorem to show that for a >= 6, the only obstruction to embedding an ellipsoid E(1,a) into a scaling of E(1,3/2) is the volume, and we also give new proofs of similar results for embeddings into scalings of E(1,1) and E(1,2).

24 pages, with 1 figure

References in corpus (7)

• Symplectic embeddings of 4-dimensional ellipsoids: erratum
• Maximal Periods of (Ehrhart) Quasi-Polynomials
• Symplectic embeddings of 4-dimensional ellipsoids into cubes
• The ghost stairs stabilize to sharp symplectic embedding obstructions
• Infinite staircases in the symplectic embedding problem for four-dimensional ellipsoids into polydisks
• Sub-leading asymptotics of ECH capacities
• Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Cited by in corpus (13)

• Beyond ECH capacities
• When symplectic topology meets Banach space geometry
• Symplectic embeddings of four-dimensional ellipsoids into integral polydiscs
• Sub-leading asymptotics of ECH capacities
• ECH capacities, Ehrhart theory, and toric varieties
• New examples of period collapse
• Special eccentricities of rational four-dimensional ellipsoids
• ECH embedding obstructions for rational surfaces
• Symplectic embeddings of products
• Towers of Looijenga pairs and asymptotics of ECH capacities
• Higher Symplectic Capacities and the Stabilized Embedding Problem for Integral Ellipsoids
• New irrational polygons with Ehrhart-theoretic period collapse
• Newton-Okoukov bodies and symplectic embeddings into non-toric rational surfaces