Ewald's Conjecture and integer points in algebraic and symplectic toric geometry
Abstract
We solve several open problems concerning integer points of polytopes arising in symplectic and algebraic geometry. In this direction we give the first proof of a broad case of Ewald's Conjecture (1988) concerning symmetric integral points of monotone lattice polytopes in arbitrary dimension. We also include an asymptotic quantitative study of the set of points appearing in Ewald's Conjecture. Then we relate this work to the problem of displaceability of orbits in symplectic toric geometry. We conclude with a proof for the -dimensional case, and for a number of cases in higher dimensions, of Nill's Conjecture (2009), which is a generalization of Ewald's conjecture to smooth lattice polytopes. Along the way the paper introduces two new classes of polytopes which arise naturally in the study of Ewald's Conjecture and symplectic displaceability: neat polytopes, which are related to Oda's Conjecture, and deeply monotone polytopes.
Cite
@article{arxiv.2310.10366,
title = {Ewald's Conjecture and integer points in algebraic and symplectic toric geometry},
author = {Luis Crespo and Álvaro Pelayo and Francisco Santos},
journal= {arXiv preprint arXiv:2310.10366},
year = {2026}
}
Comments
37 pages, 6 figures. Added third co-author as a result of improvements. In particular we strenghten our previous theorem on Ewald's Conjecture by introducing deeply mononote polytopes and prove new cases of Nill's Conjecture