Most (0,1)-polytopes are not normal

@misc{morales2026most,
  title = {{Most $(0,1)$-polytopes are not normal}},
  author = {Santiago Morales},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02778},
  eprint = {2609.02778},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02778},
}

TY  - UNPB
TI  - Most (0,1)-polytopes are not normal
AU  - Santiago Morales
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02778
UR  - https://arxiv.org/abs/2609.02778
AB  - We prove that the proportion of 0/1-equivalence classes of d-dimensional (0,1)-polytopes that are normal tends to zero at least at a double exponential rate as d→∞. As a consequence, the same holds for any of the following classes given by the type of triangulation possible: (a) quadratic, (b) flag unimodular, (c) regular unimodular, or (d) unimodular, among others. We classify the 0/1-equivalence classes of d-dimensional (0,1)-polytopes for d≤5 according to whether they admit a unimodular, flag unimodular, or quadratic triangulation. In dimension five, exactly $175$ out of 1,226,525 classes have a flag unimodular triangulation, but no quadratic triangulation. Among them, there are polytopes whose toric rings are not Koszul; thus, we find the first polytopes that have a flag unimodular triangulation, but whose toric ring is not Koszul. In contrast with the matroid case, we exhibit a delta-matroid polytope that is not normal.
ER  -