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arxiv: 2310.10366 · v3 · submitted 2023-10-16 · 🧮 math.CO · math.AG· math.SG

Ewald's Conjecture and integer points in algebraic and symplectic toric geometry

Luis Crespo , \'Alvaro Pelayo , Francisco Santos This is my paper

Pith reviewed 2026-05-24 06:25 UTC · model grok-4.3

classification 🧮 math.CO math.AGmath.SG
keywords Ewald's conjecturemonotone lattice polytopessymmetric integral pointsNill's conjecturetoric geometrydisplaceabilityneat polytopesdeeply monotone polytopes
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The pith

A broad case of Ewald's 1988 conjecture holds: monotone lattice polytopes possess symmetric integral points in every dimension.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper delivers the first proof that monotone lattice polytopes contain symmetric integral points across arbitrary dimensions, settling a substantial portion of Ewald's conjecture. It supports this with an asymptotic count of the points and links the result to questions of orbit displaceability in symplectic toric geometry. The authors also verify Nill's conjecture in the two-dimensional case and several higher-dimensional instances while defining two new families of polytopes that arise naturally in the argument. A reader would care because these statements constrain the integer-point structure of polytopes that appear in both algebraic and symplectic constructions. The proofs rely on combinatorial properties rather than dimension-specific geometry.

Core claim

Every monotone lattice polytope admits symmetric integral points, at least under the broad combinatorial conditions treated here, and the set of such points admits an asymptotic quantitative description. This statement is proved in arbitrary dimension. The same framework yields a proof of Nill's conjecture for smooth lattice polytopes in dimension two and selected cases in higher dimensions, together with relations between the points and the displaceability of orbits in the associated toric manifolds.

What carries the argument

The combinatorial definition of monotone lattice polytopes together with the precise notion of symmetric integral points, which extends the argument beyond low dimensions.

If this is right

  • Symmetric integral points exist for all monotone lattice polytopes in any dimension under the combinatorial conditions used.
  • An asymptotic formula describes the growth of the set of points satisfying the conditions of Ewald's conjecture.
  • Nill's conjecture holds in dimension two and in several families of higher-dimensional smooth lattice polytopes.
  • The new classes of neat polytopes and deeply monotone polytopes organize the cases that relate to Oda's conjecture and symplectic displaceability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The introduction of neat polytopes may supply a route toward resolving Oda's conjecture if their relation to toric varieties can be made fully explicit.
  • Displaceability results in symplectic toric manifolds could be tested directly by constructing the moment maps of the newly defined deeply monotone polytopes.
  • The asymptotic count of symmetric points might be refined to give exact formulas in fixed dimension once the combinatorial types are enumerated.
  • The same combinatorial techniques could apply to other classes of lattice polytopes that arise in mirror symmetry or in the study of Fano varieties.

Load-bearing premise

The combinatorial definition of monotone lattice polytopes already encodes all geometric conditions required for the existence statement to hold in arbitrary dimension.

What would settle it

A single explicit monotone lattice polytope of dimension four or higher that contains no pair of symmetric integral points would refute the broad case proved here.

Figures

Figures reproduced from arXiv: 2310.10366 by \'Alvaro Pelayo, Francisco Santos, Luis Crespo.

Figure 1
Figure 1. Figure 1: The five 2-dimensional monotone polygons. We call them the monotone triangle, trapezoid, square, pentagon, and hexagon, respectively Recall that a rational polytope is called smooth if it is simple and every normal cone is unimodular, and reflexive if it is a lattice polytope with the origin in its interior and its polar is also a lattice polytope. Smooth polytopes in general, and monotone ones in particul… view at source ↗
Figure 2
Figure 2. Figure 2: The first two figures show unimodular bases B1 and B2 (yellow points) of two hyperplanes (blue), as obtained in the proof of Lemma 4.7 for the case of the 3-cube, where F is the facet pointing forward. The third figure shows the resulting unimodular basis of Z 3 . Corollary 4.8. All 8-dimensional UT-free monotone polytopes satisfy the strong Ewald Condition. More generally, if all monotone polytopes in dim… view at source ↗
Figure 3
Figure 3. Figure 3: The only two non-UT-free 3-dimensional monotone polytopes. dimension monotone monotone UT-free deeply monotone 3 18 16 16 4 124 74 72 5 866 336 300 6 7622 1699 1352 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Two monotone bundles. The first has a segment as base and a square as fiber. The second has a hexagon as base and a segment as fiber. Example 5.3. A Cartesian product is a bundle, where any one of the two factors is the base and the other the fiber. See [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: From left to right: SSB(3, 0), SSB(3, 1), SSB(3, 2). n − k ⩾ n − 1, that is, k ⩽ 1. Sufficiency follows by induction on n, observing that the first displacements of the facets of SSB(n, k), for k ⩽ 1, are the monotone simplex (for the facets that are simplices) and the rest give a monotone segment, if n = 2, and SSB(n − 1, k), if n ⩾ 3 and k < n − 1. □ We can also check the Ewald properties for these polyt… view at source ↗
Figure 6
Figure 6. Figure 6: The momentum polytopes of the complex projective space CP 2 and of CP 1 ×CP 1 are a monotone triangle (left) and square (right). The toric fiber of a point in the interior is a 2-torus (S 1 ) 2 , that of a point along an edge is a “1-torus” (that is, a circle) and that of a vertex is a point. See Example 8.2. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_6.png] view at source ↗
read the original 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 $2$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to give the first proof of a broad case of Ewald's 1988 conjecture on the existence of symmetric integral points for monotone lattice polytopes in arbitrary dimension. It also provides an asymptotic quantitative study of such points, relates the results to orbit displaceability in symplectic toric geometry, proves Nill's 2009 conjecture in dimension 2 and selected higher-dimensional cases, and introduces the auxiliary classes of neat polytopes and deeply monotone polytopes.

Significance. If the central proof is correct, the result would resolve a long-standing open problem at the interface of combinatorial convex geometry and toric algebraic/symplectic geometry, with potential implications for questions on lattice-point symmetry and Hamiltonian displaceability. The asymptotic analysis and new polytope classes could serve as tools for further work in the area.

major comments (2)
  1. [Introduction, second paragraph; proof section for Ewald's conjecture] Introduction (second paragraph) and the section containing the proof of the broad case of Ewald's conjecture: the argument proceeds from the combinatorial definition of monotone lattice polytopes together with the notion of symmetric integral points, yet the manuscript must explicitly verify that these definitions encode all relevant geometric conditions (e.g., origin monotonicity in the toric variety and conditions for displaceability) that could produce obstructions only in dimension ≥3; the introduction of the auxiliary classes 'neat polytopes' and 'deeply monotone polytopes' indicates that the base combinatorial data may not be sufficient by itself.
  2. [Definition of deeply monotone polytopes] Section defining deeply monotone polytopes: the geometric content of this new class (and its relation to the toric/symplectic setting) is not shown to be fully reducible to the combinatorial data used for the main theorem; a counter-example in which the combinatorial conditions hold but a geometric obstruction appears would falsify the extension to arbitrary dimension.
minor comments (2)
  1. [Abstract] The abstract states that the paper 'relates this work to the problem of displaceability of orbits' but does not indicate which specific theorem or corollary establishes the relation; a forward reference would improve clarity.
  2. [Throughout] Notation for the new classes (neat polytopes, deeply monotone polytopes) should be introduced with a single consistent symbol or abbreviation on first use to avoid repeated full-phrase repetition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need to make the correspondence between combinatorial definitions and geometric conditions fully explicit. We agree that clarifications are warranted and will revise the manuscript accordingly to strengthen the exposition without altering the core arguments.

read point-by-point responses

  1. Referee: Introduction (second paragraph) and the section containing the proof of the broad case of Ewald's conjecture: the argument proceeds from the combinatorial definition of monotone lattice polytopes together with the notion of symmetric integral points, yet the manuscript must explicitly verify that these definitions encode all relevant geometric conditions (e.g., origin monotonicity in the toric variety and conditions for displaceability) that could produce obstructions only in dimension ≥3; the introduction of the auxiliary classes 'neat polytopes' and 'deeply monotone polytopes' indicates that the base combinatorial data may not be sufficient by itself.

    Authors: The combinatorial definitions of monotone lattice polytopes and symmetric integral points are chosen to encode the geometric conditions of origin monotonicity and orbit displaceability in the toric setting, as these are standard translations in the literature (e.g., via the correspondence between lattice polytopes and toric varieties). The auxiliary classes of neat and deeply monotone polytopes are introduced precisely to handle potential higher-dimensional obstructions by imposing additional combinatorial conditions that ensure the geometric properties hold. We will add an explicit paragraph or short subsection in the introduction and proof section verifying this encoding and referencing the relevant toric geometry background to address any ambiguity. revision: yes

  2. Referee: Section defining deeply monotone polytopes: the geometric content of this new class (and its relation to the toric/symplectic setting) is not shown to be fully reducible to the combinatorial data used for the main theorem; a counter-example in which the combinatorial conditions hold but a geometric obstruction appears would falsify the extension to arbitrary dimension.

    Authors: The definition of deeply monotone polytopes is combinatorial but is constructed so that it directly implies the required geometric properties (no obstructions to symmetric points or displaceability) in the toric/symplectic context; the main theorem's proof demonstrates that these conditions suffice for the broad case in arbitrary dimension. We will add a lemma or explicit statement in the relevant section showing the implication from the combinatorial definition to the absence of geometric obstructions, thereby confirming reducibility and ruling out the possibility of counterexamples under the stated hypotheses. revision: yes

Circularity Check

0 steps flagged

No circularity; proof rests on external conjecture and combinatorial definitions

full rationale

The paper claims a first proof of a case of Ewald's 1988 conjecture for symmetric integral points of monotone lattice polytopes, proceeding from the combinatorial definition of those polytopes together with the notion of symmetric integral points. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the provided text. New classes (neat polytopes, deeply monotone polytopes) are introduced as auxiliary tools rather than used to define the target result. The derivation chain is therefore independent of its own outputs and does not reduce by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claims rest on standard definitions of lattice polytopes, monotonicity, and toric geometry from prior literature; the paper adds two new classes of polytopes whose properties are used in the proofs.

axioms (1)
  • standard math Standard axioms and definitions of lattice polytopes, monotone polytopes, and toric varieties from algebraic and symplectic geometry.
    Invoked throughout to set up Ewald's and Nill's conjectures.
invented entities (2)
  • neat polytopes no independent evidence
    purpose: New class introduced to study Ewald's conjecture and related to Oda's conjecture.
    Defined in the paper as a tool for the proofs.
  • deeply monotone polytopes no independent evidence
    purpose: New class arising in the study of Ewald's conjecture and symplectic displaceability.
    Defined in the paper as a tool for the proofs.

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Reference graph

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