{
  "abstract": "We prove that deciding whether an ideal in a polynomial ring contains a trinomial is impossible on a Turing machine. More precisely, from an integer polynomial $P$ we compute generators of an ideal $I_P$ in a polynomial ring over $\\mathbb{Q}$ such that $I_P$ contains a trinomial if and only if $P$ has an integral zero. By the MRDP theorem this problem is undecidable. A universal halting polynomial gives a computable family of ideals in one fixed polynomial ring, with uniform bounds on colength, generator count, and generator degree, for which the containment of a trinomial encodes the halting problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.31162",
  "author_total": 3,
  "authors": [
    {
      "name": "Tobias Boege"
    },
    {
      "name": "Anna Hofer"
    },
    {
      "name": "Thomas Kahle"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8k77xb71m3bas.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.31162",
  "fulltext_url": null,
  "id": "8k77xb71m3bas",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.31162",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AC"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7204952146",
  "pdf_url": "https://arxiv.org/pdf/2608.31162",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8k77xb71m3bas.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.31162",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.31162"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AC",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Trinomial containment in polynomial ideals is undecidable",
  "title_text": "Trinomial containment in polynomial ideals is undecidable",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/8k77xb71m3bas",
  "venue": "arXiv",
  "versions": []
}