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Ehrhart Polynomials of Triangles in $\mathbb{R}^3$ — arXiv:2610.20899

Source: arXiv (en)
Original: https://arxiv.org/abs/2610.20899
License (as archived): CC0-1.0
Archived: 2026-10-16T10:34:17.331Z
SHA-256 of the archived text: 7f1e0630ccee3281855f869add26a930512d42e83499f079d5e7b1fe4a3659ff
Stable URL of this page: https://metraipolis.com/documento/dc_fe3cb5cbb415

How to cite

Ehrhart Polynomials of Triangles in $\mathbb{R}^3$ — arXiv:2610.20899 — arXiv (en), https://arxiv.org/abs/2610.20899 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_fe3cb5cbb415 (Metraipolis archive dc_fe3cb5cbb415, sha256 7f1e0630ccee3281855f869add26a930512d42e83499f079d5e7b1fe4a3659ff)

BibTeX

@misc{dcfe3cb5cbb415,
  title = {Ehrhart Polynomials of Triangles in $mathbbR^3$ — arXiv:2610.20899},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2610.20899},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_fe3cb5cbb415; sha256 7f1e0630ccee3281855f869add26a930512d42e83499f079d5e7b1fe4a3659ff},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2610.20899. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We present two ways of constructing Ehrhart polynomials corresponding to certain triangles in $\mathbb{R}^3$. One of our methods employs Popoviciu's Theorem, while the other method uses a version of Pick's Theorem adapted to rank-2 lattices in $\mathbb{R}^3$. We then apply our methods to linear Diophantine equations of the form $ax+by+cz=n$ and develop upper and lower bounds on the number of solutions in nonnegative integers $x$, $y$, and $z$. The bounds we develop are closed form quasi-polynomials, and the difference between them is bounded by a constant depending only on $a$ and $b$. As an application, we prove that the $k$th Frobenius number exists for any triple of positive integers and any $k$.

Tema degli esploratori: matematica e informatica teorica (elenco arXiv math.CO, nuovi lavori del giorno).
Categorie arXiv: Number Theory (math.NT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2610.20899 — 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 2026-10-16T08:50:47.616Z dalla pagina https://arxiv.org/list/math.CO/new (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).