{
  "abstract": "To a polycube $\\mathcal P$ in $\\mathbb R^n$, we associate an ideal $I_{\\mathcal P}$ generated by the join--meet relations of vertices that are corners of a cuboid of $\\mathcal P$. For $n=2$ these are the polyomino ideals. In contrast with the planar case, we show that for $n\\geq 3$ the ideal $I_{\\mathcal P}$ is prime if and only if $\\mathcal P$ is a cuboid. For convex polycubes in $\\mathbb R^3$ of uniform thickness whose vertex set is a sublattice of $\\mathbb N^3$, the defining relations form a quadratic Gröbner basis, and we characterize the Cohen--Macaulay ones in terms of a poset on the vertices of $\\mathcal P$; in this case the order complex of the poset is vertex decomposable. Finally, as an application, we show that the polycube algebra of a thickened $L$-convex polyomino is Koszul, and we characterize when it is Cohen--Macaulay.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12191",
  "author_total": 2,
  "authors": [
    {
      "name": "Ayesha Asloob Qureshi"
    },
    {
      "name": "Dharm Veer"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fndzfvffg5ap2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12191",
  "fulltext_url": null,
  "id": "fndzfvffg5ap2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12191",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7221077599",
  "pdf_url": "https://arxiv.org/pdf/2610.12191",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fndzfvffg5ap2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.12191",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12191"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Polycubes and polycube ideals",
  "title_text": "Polycubes and polycube ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/fndzfvffg5ap2",
  "venue": "arXiv",
  "versions": []
}