{
  "abstract": "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 $k$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20899",
  "author_total": 3,
  "authors": [
    {
      "name": "Max Aldrich"
    },
    {
      "name": "Daniel Gulbrandsen"
    },
    {
      "name": "Clara Keddy"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a7gty5j0eb50x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20899",
  "fulltext_url": null,
  "id": "a7gty5j0eb50x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20899",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7222687070",
  "pdf_url": "https://arxiv.org/pdf/2610.20899",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a7gty5j0eb50x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.20899",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20899"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Enumerating Integral Points on Triangles in $\\mathbb{R}^3$",
  "title_text": "Enumerating Integral Points on Triangles in ℝ^3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:31Z",
  "url": "https://research.draconitic.com/paper/a7gty5j0eb50x",
  "venue": "arXiv",
  "versions": []
}