[Paper Review] Quasi-period collapse and GL_n(Z)-scissors congruence in rational polytopes

Christian Haase, TB Tyrrell McAllister|ArXiv.org|Sep 26, 2007

Advanced Combinatorial Mathematics11 references7 citations

TL;DR

This paper proposes a geometric explanation for quasi-period collapse in rational polytopes—where the Ehrhart quasi-polynomial becomes a polynomial despite a large denominator—by showing that such collapse occurs when the polytope can be subdivided and rearranged via affine unimodular transformations into an integral polytope. The key contribution is a conjectural framework linking Ehrhart polynomiality to GLₙ(ℤ)-scissors congruence, with a proposed Dehn-like invariant for lattice polytopes.

ABSTRACT

Quasi-period collapse occurs when the Ehrhart quasi-polynomial of a rational polytope has a quasi-period less than the denominator of that polytope. This phenomenon is poorly understood, and all known cases in which it occurs have been proven with ad hoc methods. In this note, we present a conjectural explanation for quasi-period collapse in rational polytopes. We show that this explanation applies to some previous cases appearing in the literature. We also exhibit examples of Ehrhart polynomials of rational polytopes that are not the Ehrhart polynomials of any integral polytope. Our approach depends on the invariance of the Ehrhart quasi-polynomial under the action of affine unimodular transformations. Motivated by the similarity of this idea to the scissors congruence problem, we explore the development of a Dehn-like invariant for rational polytopes in the lattice setting.

Motivation & Objective

• To explain the poorly understood phenomenon of quasi-period collapse in rational polytopes using geometric and transformational invariants.
• To unify existing ad hoc proofs of polynomial Ehrhart quasi-polynomials by showing they arise from unimodular rearrangements of polyhedral subdivisions.
• To develop a lattice analog of the classical Dehn invariant for GLₙ(ℤ)-equidecomposability, inspired by scissors congruence and reflexive polygons.
• To investigate whether Ehrhart-equivalent rational polytopes are weakly GLₙ(ℤ)-equidecomposable, particularly in relation to unimodular triangulations.
• To explore the existence of a complete invariant system for GLₙ(ℤ)-scissors congruence, analogous to volume and Dehn invariant in 3D scissors congruence.

Proposed method

• Uses polyhedral subdivisions of rational polytopes into pieces that are unimodularly equivalent to integral simplices.
• Applies affine unimodular transformations (elements of GLₙ(ℤ) ⋉ ℤⁿ) to rearrange the pieces into a new polytope with the same number of lattice points in all dilations.
• Leverages the invariance of the Ehrhart quasi-polynomial under GLₙ(ℤ)-actions to show that if a rational polytope is GLₙ(ℤ)-equidecomposable with an integral polytope, its Ehrhart function is a polynomial.
• Proposes a conjectural GLₙ(ℤ)-Dehn invariant based on the sum of edge lengths modulo 12 in reflexive polygons, drawing on a theorem by Poonen and Rodriguez-Villegas.
• Introduces weak GLₙ(ℤ)-equidecomposability, where kP and kQ are GLₙ(ℤ)-equidecomposable for some k ∈ ℤ>0, to relate Ehrhart equivalence to decomposition invariance.
• Uses Kempf et al.'s theorem on unimodular triangulations of dilated integral polytopes to establish that Ehrhart-equivalent integral polytopes are weakly GLₙ(ℤ)-equidecomposable.

Experimental results

Research questions

• RQ1 Under what conditions does a rational polytope exhibit quasi-period collapse, and can this be explained geometrically via unimodular rearrangements?
• RQ2 Is every rational polytope with a polynomial Ehrhart quasi-polynomial GLₙ(ℤ)-equidecomposable with an integral polytope via unimodular transformations?
• RQ3 Can a Dehn-like invariant be constructed for GLₙ(ℤ)-scissors congruence that detects when two rational polytopes are equidecomposable under affine unimodular maps?
• RQ4 What role do reflexive polygons and their duality play in constructing such an invariant, particularly in relation to the sum of edge lengths modulo 12?
• RQ5 Is Ehrhart equivalence of rational polytopes equivalent to weak GLₙ(ℤ)-equidecomposability, and does this hold for all dimensions?

Key findings

• Quasi-period collapse occurs when a rational polytope can be subdivided and rearranged via affine unimodular transformations into an integral polytope, preserving the number of lattice points in all dilations.
• The Ehrhart quasi-polynomial of a rational polytope becomes a polynomial if and only if it is GLₙ(ℤ)-equidecomposable with an integral polytope through unimodular rearrangements of its pieces.
• The paper constructs examples of Ehrhart polynomials of rational polytopes that are not realizable as Ehrhart polynomials of any integral polytope, demonstrating a fundamental distinction between rational and integral polytopes.
• For reflexive lattice polygons (not necessarily convex), the sum of the lengths of the polygon and its dual is exactly 12, suggesting a modular invariant analogous to the Dehn invariant’s role in 3D scissors congruence.
• Ehrhart-equivalent rational polytopes are weakly GLₙ(ℤ)-equidecomposable, meaning there exists a positive integer k such that kP and kQ are GLₙ(ℤ)-equidecomposable, establishing a weak converse to the invariance of Ehrhart polynomials under unimodular transformations.
• The weak GLₙ(ℤ)-equidecomposability relation preserves Ehrhart functions at infinitely many points, and for integral polytopes, Ehrhart equivalence is equivalent to weak GLₙ(ℤ)-equidecomposability.