{
  "abstract": "We study the structure and distribution of atoms in the monoid $\\mathcal{M}{\\rm on}(R)$ of nonzero monomial ideals of the polynomial ring $R=K[X_1,\\ldots, X_N]$, with $N\\ge 2$. We characterize the atoms with at most four minimal generators and introduce the density $d(N,μ)$ of atoms among monomial ideals with $μ$ minimal generators. For $μ\\le 4$, we prove the existence of this density and determine its asymptotic behavior as $N\\to\\infty$. In the bivariate case, we also obtain a uniform lower bound for the density of atoms in a divisor-closed submonoid of $\\mathcal{M}{\\rm on}(R)$ generated by equigenerated ideals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17202",
  "author_total": 2,
  "authors": [
    {
      "name": "Nikola Bogdanovic"
    },
    {
      "name": "Laura Cossu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9sezsyrjf2yvc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17202",
  "fulltext_url": null,
  "id": "9sezsyrjf2yvc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17202",
  "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.AC"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213435573",
  "pdf_url": "https://arxiv.org/pdf/2609.17202",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9sezsyrjf2yvc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17202",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17202"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "On atoms and their density in the monoid of monomial ideals",
  "title_text": "On atoms and their density in the monoid of monomial ideals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/9sezsyrjf2yvc",
  "venue": "arXiv",
  "versions": []
}