Non-Unimodal Ehrhart Polynomials of Reflexive Simplices

Definitions and mathematical statement

Use the standard lattice interpretation of the displayed family. For an integer $n\geq 1$, let $e_1,\ldots,e_n$ be the standard basis of ${\mathbb{R}}^n$, and let $q=(q_1,\ldots,q_n)\in{\mathbb{Z}}^n$. Set \[ \Delta_{(1,q)}=\operatorname{conv}\{e_1,\ldots,e_n,-q\}. \] The source does not explicitly specify the parameter domain; the interpretation here is that these are full-dimensional lattice simplices in ${\mathbb{Z}}^n$, with reflexivity taken about the origin. In this interpretation, the requirement that the origin lie in the interior forces $q_i>0$ for every $i$.

A full-dimensional lattice polytope $P\subset{\mathbb{R}}^n$ is reflexive if $0\in\operatorname{int}(P)$ and its polar \[ P^\vee=\{y\in({\mathbb{R}}^n)^*: \langle y,x\rangle\leq 1\text{ for every }x\in P\} \] is a lattice polytope in the dual lattice. It has the integer-decomposition property (IDP) if, for every integer $k\geq 1$ and every $z\in kP\cap{\mathbb{Z}}^n$, there exist $z_1,\ldots,z_k\in P\cap{\mathbb{Z}}^n$ such that $z=z_1+\cdots+z_k$.

Define the Ehrhart $h^*$-polynomial by \[ \sum_{m=0}^{\infty}\#(mP\cap{\mathbb{Z}}^n)t^m =\frac{h_P^*(t)}{(1-t)^{n+1}}, \qquad h_P^*(t)=\sum_{i=0}^{n}h_i^*t^i. \] Its coefficient sequence is unimodal if there is an index $r\in\{0,\ldots,n\}$ with \[ h_0^*\leq\cdots\leq h_r^*\geq\cdots\geq h_n^*. \] It is nonunimodal if no such index exists.

Classification question in the supplied clean statement. Characterize all pairs $(n,q)$, with $n\geq1$ and $q\in{\mathbb{Z}}_{>0}^n$, for which $\Delta_{(1,q)}$ is reflexive, has IDP, and has a nonunimodal Ehrhart $h^*$-polynomial.

Unresolved qualification concerning the original question. The original wording asks how likely nonunimodality is among the reflexive IDP members of this family. It specifies neither a probability measure nor a finite sampling family or limiting procedure, and does not say whether dimension is fixed. Consequently, a numerical probability or asymptotic frequency is not defined by the supplied record. The classification question above records the clean formulation; it does not resolve the missing choices in the original frequency question.

Short English statement

Determine which reflexive lattice simplices $\Delta_{(1,q)}$ with the integer-decomposition property have Ehrhart $h^*$-coefficients that fail to be unimodal. The original request for a likelihood also requires an unspecified sampling rule or limiting procedure. The supplied partially solved label is dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $30003598$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied. The record gives proposed year 2017 and labels the clean statement corrected_verified; that label reports supplied review metadata, not independent verification here.
• Supplied primary provenance: Mini-Workshop: Lattice Polytopes: Methods, Advances, Applications (2018), Oberwolfach Reports, pp. 2659--2701, DOI 10.4171/owr/2017/44. https://doi.org/10.4171/owr/2017/44. The background describes the reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Braun--Davis--Solus, reflexive simplices. https://doi.org/10.1016/j.aam.2018.06.003.
• Supplied literature reference: Braun et al., weighted projective space simplices. https://arxiv.org/abs/2103.17156.
• Supplied literature reference: original OWR report. https://doi.org/10.4171/OWR/2017/44.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. The record labels the problem partially solved and reports that explicit formulas and broad families of $\Delta_{(1,q)}$ have proved $h^*$-unimodality, that reflexive stabilizations provide nonunimodal examples outside the IDP regime, and that no theorem determines the frequency or general avoidance question for the full family. The background expressly calls this a dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.