Let P be a d-dimensional lattice simplex with the integer decomposition property: for every integer m≥ 1 and every x∈ mP capZ^d, there exist x_1,…,x_m∈ P capZ^d such that x=x_1+…+x_m. Write its Ehrhart series as ∑_m≥ 0|mP capZ^d|t^m=frach_P^(t)(1-t)^d+1, where h_P^(t)=∑_i=0^d h_i^t^i. Is the coefficient sequence (h_0^,h_1^,…,h_d^) unimodal for every such simplex; that is, does there exist an index k with h_0^≤…≤ h_k^≥…≥ h_d^?

Unimodality conjecture for h*-polynomials of IDP lattice simplices

Let $P$ be a $d$ -dimensional lattice simplex with the integer decomposition property: for every integer $m\ge 1$ and every $x\in mP\cap\mathbb{Z}^d$ , there exist $x_1,\ldots,x_m\in P\cap\mathbb{Z}^d$ such that $x=x_1+\cdots+x_m$ . Write its Ehrhart series as $\sum_{m\ge 0}|mP\cap\mathbb{Z}^d|t^m=\frac{h_P^*(t)}{(1-t)^{d+1}}$ , where $h_P^*(t)=\sum_{i=0}^d h_i^*t^i$ . Is the coefficient sequence $(h_0^*,h_1^*,\ldots,h_d^*)$ unimodal for every such simplex; that is, does there exist an index $k$ with $h_0^*\le\cdots\le h_k^*\ge\cdots\ge h_d^*$ ?

References

Primary source

arXiv

Additional references

Unimodality for IDP Lattice Simplices of Prime Normalized Volume — arXiv — Feihu Liu, Zihao Zhang

ehrhart-theory polyhedral-geometry

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed progress

The conjecture remains open in general, but a September 2026 preprint proves it for simplices whose normalized volume is prime and for some further subclasses.

The conjecture asks whether every IDP lattice simplex has a unimodal coefficient sequence in its $h^\ast$ -polynomial. The retrieved sources do not identify the original proposer or date.

Known results

Adiprasito, Papadakis, Petrotou, and Steinmeyer (2022) proved unimodality for Gorenstein IDP polytopes, including reflexive IDP lattice simplices.

September 2026 prime-volume result

Feihu Liu and Zihao Zhang’s preprint claims unimodality for every IDP lattice simplex of prime normalized volume, along with additional sufficient conditions. This is progress on, but not a resolution of, the full conjecture.

Current status (as of September 2026): Unimodality is established for reflexive IDP simplices, prime-normalized-volume IDP simplices, and other stated subclasses, but remains open for arbitrary IDP lattice simplices.

Sources

No solutions have been posted yet.