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The Hibi--Li (2016) conjecture is fully resolved — arXiv:2610.40703

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.40703
License (as archived): CC0-1.0
Archived: 2026-11-03T08:17:55.438Z
SHA-256 of the archived text: 02569b05564b1803fecd3d08901a7ce66ffd929d6d9d704c1bc103c0215bde44
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The Hibi--Li (2016) conjecture is fully resolved — arXiv:2610.40703 — arXiv (en), https://arxiv.org/abs/2610.40703 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_3cd2cc4bc601 (Metraipolis archive dc_3cd2cc4bc601, sha256 02569b05564b1803fecd3d08901a7ce66ffd929d6d9d704c1bc103c0215bde44)

BibTeX

@misc{dc3cd2cc4bc601,
  title = {The Hibi--Li (2016) conjecture is fully resolved — arXiv:2610.40703},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.40703},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_3cd2cc4bc601; sha256 02569b05564b1803fecd3d08901a7ce66ffd929d6d9d704c1bc103c0215bde44},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.40703. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

Hibi and Li conjectured that the order polytope $\mathcal{O}(P)$ of a finite poset $P$ has at most as many faces of each dimension as its chain polytope $\mathcal{C}(P)$, and that equality in some dimension $k$ with $1\leq k\leq|P|-1$ forces the two polytopes to be unimodularly equivalent. Fourier proved the inequality in a 2026 preprint. Let $X$ be the five-element poset in which an element lies above two incomparable elements and below two other incomparable elements. Building on Fourier's proof, we show that if an $n$-element poset contains $X$ as an induced subposet, then, for every $k$ with $2\leq k\leq n-1$, its chain polytope has at least $\binom{n-3}{k-2}$ more $k$-dimensional faces than its order polytope, and that for every $n\geq5$ some $n$-element poset containing $X$ attains all these bounds at once. Since Hibi and Li showed that $\mathcal{O}(P)$ and $\mathcal{C}(P)$ are unimodularly equivalent exactly when $P$ does not contain $X$, this proves the equality statement for these $k$. For $k=1$ the equality statement is false: $\mathcal{O}(X)$ and $\mathcal{C}(X)$ have equally many edges (Hibi, Li, Sahara and Shikama) but are not unimodularly equivalent. With Fourier's inequality, these results settle the conjecture. Every result is formally verified in Lean 4, with only published results and two results of Fourier's preprint as axioms.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Combinatorics (math.CO) ; Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2610.40703 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-11-03T08:05:23.764Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).