{
  "abstract": "A monomial square form is a polynomial $F=\\sum_{i=1}^k M_i^2$, where $M_1,\\ldots,M_k$ are pairwise distinct squarefree monomials $\\ne1$. We consider the problem of which of these forms are universal, i.e.\\ represent every positive integer. We consider three classes of such forms:\\\\ (1) Forms $M_1^2+M_2^2+M_3^2+M_4^2$ ($k=4$),\\\\ (2) Forms $M_1^2+\\ldots +M_5^2$ ($k=5$) in at most 5 variables,\\\\ (3) Forms where the monomials are products of consecutive variables of a fixed length:\\\\ $P_{m,d}(x)=\\sum_{i=1}^{d}(x_ix_{i+1}\\cdots x_{i+m-1})^2,$ and its circular version\\\\ $C_{m,d}(x)=\\sum_{i=1}^{n}(x_ix_{i+1}\\cdots x_{i+m-1})^2\\quad(n=m+d-1,\\ x_{i+n}=x_i).$ We classify which of these are universal, possibly universal, or conditionally universal on a conjecture on the form $x^2+4y^2+5z^2$. This ternary form appears repeatedly and seems to play a central role. For forms (1), we also show that a form is universal if and only if it represents 14, 15, 29 and 9591 (partially conditional on $x^2+4y^2+5z^2$).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.03057",
  "author_total": 1,
  "authors": [
    {
      "name": "Stefan Maubach"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1h5mtd40axa79.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.03057",
  "fulltext_url": null,
  "id": "1h5mtd40axa79",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.03057",
  "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.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.03057",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1h5mtd40axa79.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:14Z",
      "native_id": "2611.03057",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.03057"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Universality of monomial square forms",
  "title_text": "Universality of monomial square forms",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/1h5mtd40axa79",
  "venue": "arXiv",
  "versions": []
}