γ-positivity for duals of symmetric edge polytopes
The h*-polynomials of symmetric edge polytopes are not always γ-positive (counterexample, 2026). What about their polar duals, a class of alcoved polytopes?
Cite
@misc{cairn-alcoved-polytopes-gamma-positivity,
title = {γ-positivity for duals of symmetric edge polytopes},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/alcoved-polytopes-gamma-positivity}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
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The problem
The question
Symmetric edge polytopes of graphs are reflexive lattice polytopes; their polar duals form a subclass of alcoved polytopes. Study the Ehrhart theory of these duals; in particular, are their h*-polynomials γ-positive? (A palindromic polynomial h(x) of degree d is γ-positive if h(x) = Σ γ_i x^i (1 + x)^{d − 2i} with all γ_i ≥ 0.)
What is known
The Ohsugi–Tsuchiya conjecture that h*-polynomials of symmetric edge polytopes themselves are γ-positive was disproved in 2026 with series-parallel counterexamples. The question for the polar duals is open.
What counts as progress
- Computation of h*-polynomials and γ-vectors of the duals for all graphs up to a given size (Normaliz, LattE), with code and data.
- A counterexample: one graph whose dual polytope has a negative γ_i — an exact, checkable computation.
- Proofs for graph families.
How it is checked
h*-vectors are exact integer computations; a negative γ_i is a certificate reproducible by independent software (level B).
Source. Posed by the participants of the mini-workshop (working-group problem) in the open problem session of the Oberwolfach workshop Mini-Workshop: Alcoved Polytopes in Physics and Optimization (2025), recorded in Oberwolfach Reports 12/2025, p. 532 (EMS Press, DOI 10.4171/OWR/2025/12), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.