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Stanley, Brenti, and Ferroni–Higashitani Ehrhart conjectures

Let P⊆RdP\subseteq\mathbb{R}^d be a dd-dimensional lattice polytope with the integer decomposition property: for every integer k≥1k\geq 1 and every x∈kP∩Zdx\in kP\cap\mathbb{Z}^d, there exist x1,…,xk∈P∩Zdx_1,\ldots,x_k\in P\cap\mathbb{Z}^d such that x=x1+⋯+xkx=x_1+\cdots+x_k. Writing the Ehrhart series as ∑m≥0∣mP∩Zd∣tm=hP∗(t)(1−t)d+1\sum_{m\geq 0}|mP\cap\mathbb{Z}^d|t^m=\frac{h_P^*(t)}{(1-t)^{d+1}}, where hP∗(t)=∑i=0shi∗tih_P^*(t)=\sum_{i=0}^s h_i^*t^i, the conjecture asserts that the coefficient sequence is unimodal: there exists an index rr such that h0∗≤⋯≤hr∗≥⋯≥hs∗h_0^*\leq\cdots\leq h_r^*\geq\cdots\geq h_s^*.

References

Primary source

arXiv

Additional references

combinatoricsehrhart-theorypolyhedral-geometry
Posted · edited

Progress summary

Refreshed
Claimed solved

An unrefereed preprint claims counterexamples to several Ehrhart coefficient-positivity conjectures, including smooth examples relevant to Stanley’s conjecture.

The entry concerns conjectures associated with Stanley, Brenti, and Ferroni–Higashitani about positivity, unimodality, and log-concavity of Ehrhart data for structured lattice polytopes. Earlier work had already produced counterexamples in broad families, but the newest claim targets Cayley sums of rectangular prisms.

Known results

  • Stanley’s order polytopes fail Ehrhart positivity in every dimension d≥14d\ge 14; explicit coefficient formulas identify the negative cases (2018).
  • Ferroni–Higashitani’s proposed unimodality for every IDP polytope has reported counterexamples to log-concavity, and hence challenges the broader positivity picture (2025).
  • A 5252-dimensional lattice polytope with non-unimodal h∗h^*-vector was constructed as a product of two polytopes with unimodal h∗h^*-vectors; a 5454-dimensional self-product example was also given (2023).

September 9, 2026 counterexamples

On September 9, 2026, the preprint Unimodality shenanigans in Ehrhart theory reported counterexamples from Cayley sums of rectangular prisms, including smooth examples for Stanley’s conjecture. The report claims negative answers to several tracked conjectures, but the preprint is unrefereed and the claim remains unverified.

Current status (as of September 2026): Several conjectures are claimed to be false via Cayley-sum counterexamples, while the new claims and their exact scope remain unverified.

Sources

Solutions 0

No solutions have been posted yet.