Regular dyadic triangulations of delta-matroid polytopes

@misc{vallee2026regular,
  title = {{Regular dyadic triangulations of delta-matroid polytopes}},
  author = {Mathieu Vallée},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18331},
  eprint = {2609.18331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18331},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Regular dyadic triangulations of delta-matroid polytopes
AU  - Mathieu Vallée
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18331
UR  - https://arxiv.org/abs/2609.18331
AB  - Backman and Liu proved that every integral generalized permutohedron of type A, and in particular every matroid base polytope, admits a regular unimodular triangulation. The analogous statement fails in type B: the delta-matroid simplex *conv{0, e_1+e_2, e_1+e_3, e_2+e_3} has normalized volume $2$ and no lattice points other than its vertices, so it has no unimodular triangulation. We show moreover that, up to the natural symmetries of the 0/1 cube and deletion of constant coordinates, it is the unique non-unimodular delta-matroid polytope that is a simplex. We prove instead that every delta-matroid polytope admits a regular dyadic triangulation, meaning a lattice triangulation whose maximal simplices have normalized volumes that are powers of two. More generally, every integral type B generalized permutohedron admits such a triangulation. The main lattice-theoretic ingredient is that the type B root configuration forms a totally dyadic system, a $2$-local analogue of total unimodularity. As a consequence, these polytopes satisfy a dyadic version of the integer decomposition property. In each dimension the corresponding exponent can be chosen uniformly, even though ordinary integer decomposition can fail for delta-matroid polytopes.
ER  -