Ehrhart unimodality for the two-row family with M=d-1

Determine whether the Ehrhart polynomial of the two-row Hermite normal form simplex associated with a=(1,...,1,N) and b=(d-2,...,d-2,d-1,0) is unimodal for every N>1.

Background

For the two-row family with M=d-1, the paper proves Ehrhart positivity and computes an explicit h*-polynomial. The authors report that computer experiments suggest unimodality of the Ehrhart polynomial for all N>1, but do not establish this analytically.

The unresolved question concerns precisely the family with b=(d-2,...,d-2,d-1,0), and asks whether the experimentally observed unimodality holds uniformly in N. This contrasts with the M=d case, where the paper proves non-unimodality for sufficiently large N.

References

For $M=d-1$, computer experiments suggest that the Ehrhart polynomial is unimodal. At present, we do not know a proof of this fact.

\begin{question}\label{question:unimodal_two_rows} Let $#1{S}{ a, b}$ be the two-row Hermite normal form simplex associated with $ a=(1,\ldots,1,N)$ and $ b=(d-2,\ldots,d-2,d-1,0)$. Is the Ehrhart polynomial of $#1{S}{ a, b}$ unimodal for all $N\inN_{>1}$? \end{question}

— Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices  (2608.28282 - Bruckamp et al., 28 Aug 2026) in Section 5, subsection “Ehrhart theoretic properties of S_{a,b},” Remark immediately preceding Question 5.1 and Question 5.1