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Sylvester simplices: Triangulations and Ehrhart-theoretic aspects — arXiv:2609.23710

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.23710
License (as archived): CC0-1.0
Archived: 2026-10-09T21:44:54.249Z
SHA-256 of the archived text: 84ba26909a51490b8c0f97b1b7481d618dd9eb83d22d483eb02ac0ee9b5970d3
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How to cite

Sylvester simplices: Triangulations and Ehrhart-theoretic aspects — arXiv:2609.23710 — arXiv (en), https://arxiv.org/abs/2609.23710 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_030dbe0f26cf (Metraipolis archive dc_030dbe0f26cf, sha256 84ba26909a51490b8c0f97b1b7481d618dd9eb83d22d483eb02ac0ee9b5970d3)

BibTeX

@misc{dc030dbe0f26cf,
  title = {Sylvester simplices: Triangulations and Ehrhart-theoretic aspects — arXiv:2609.23710},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.23710},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_030dbe0f26cf; sha256 84ba26909a51490b8c0f97b1b7481d618dd9eb83d22d483eb02ac0ee9b5970d3},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.23710. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

The Sylvester simplex $\mathsf{Sylv}_d^k$ is a $d$-dimensional lattice simplex with exactly $k$ interior lattice points. Sylvester simplices are conjectured to be the volume maximizers among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points for any $k\geq 1$. Even stronger, it is conjectured that they maximize (entry-wise) the $h^\ast$-vector among all $d$-dimensional lattice polytopes with exactly $k$ interior lattice points. Yet, Sylvester simplices seem to be rarely studied in their own right. In particular, their Ehrhart-theoretic properties are far from being well understood. In the present article, we tackle this problem. We describe flag, regular and unimodular triangulations for the Sylvester simplices, and prove that their $h^\ast$-vectors are unimodal. Moreover, we explicitly determine the values of some entries of their $f^\ast$-vectors, and prove that they are Ehrhart magic positive up to dimension $6$ but not in dimension $7$. We conclude by detailing tables of Ehrhart-theoretic quantities (numbers of lattice points, Ehrhart polynomials, local and boundary $h^\ast$-vectors, $f^\ast$-vectors) for Sylvester simplices of dimensions 7 and lower.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-22 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.23710 — 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 2026-10-09T21:26:41.129Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-22?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).