Constant-prefix one-row Hermite normal form simplices

Determine, for 1≤q<N, the values of q and N for which the one-row Hermite normal form simplex associated with a=(N-q,...,N-q,N) admits a unimodular triangulation or satisfies the integer decomposition property.

Background

The principal one-row family studied in the paper has q=1, namely a=(N-1,...,N-1,N), and its unimodular triangulation and integer decomposition properties are completely classified. The question asks whether analogous classifications can be obtained when the first d-1 entries are any common value N-q.

This would extend the paper’s results from the q=1 case to the broader class of one-row vectors with constant first d-1 coordinates. Both the existence of a unimodular triangulation and the integer decomposition property are left to be determined.

References

For which values of $q$ and $N$, with $1\leq q<N$, does the one-row Hermite normal form simplex associated with $ a=(N-q,\ldots,N-q,N)$ admit a unimodular triangulation or satisfy the integer decomposition property?

— Unimodular triangulations and Ehrhart theory for two families of Hermite normal form simplices  (2608.28282 - Bruckamp et al., 28 Aug 2026) in Section 3, immediately after the question on effective criteria