{
  "abstract": "We study triples of distinct nonconstant polynomials $a,b,c$ over $\\mathbb{R}$, $\\mathbb{Q}$ or $\\mathbb{Z}$ for which $$ ab+1,\\qquad ac+1,\\qquad bc+1,\\qquad abc+1 $$ are all polynomial squares. We show that, up to permutation and a common invertible affine change of variable, there is exactly one such triple of quadratic polynomials in $\\mathbb{R}[X]$, the one arising from a recent construction of Jurasić, and that none exists in $\\mathbb{Q}[X]$ or $\\mathbb{Z}[X]$. Moreover, for arbitrary nonconstant entries over $\\mathbb{R}$ or $\\mathbb{Q}$, we prove that the degree pattern $(2n,2m,2l)$ with $1\\le n\\le m\\le l$ satisfies either $l=m$ or $l>n+m$. For the case $l=m$, we derive factorization systems and reducibility restrictions. In the equal-degree case over $\\mathbb{Q}$, we further show that every entry has at least two distinct irreducible factors. As an application, we show that in a natural regular parametrized family of polynomial Diophantine triples in $\\mathbb{Q}[X]$ the fourth expression $abc+1$ is never a square when the parameter is constant or the first entry is a square.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05565",
  "author_total": 2,
  "authors": [
    {
      "name": "Mirela Jukić Bokun"
    },
    {
      "name": "Ana Jurasić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5snv0yvvbes95.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05565",
  "fulltext_url": null,
  "id": "5snv0yvvbes95",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05565",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7220679544",
  "pdf_url": "https://arxiv.org/pdf/2610.05565",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5snv0yvvbes95.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:10Z",
      "native_id": "2610.05565",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05565"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Four-square polynomial triples: quadratic classification and equal-degree obstructions",
  "title_text": "Four-square polynomial triples: quadratic classification and equal-degree obstructions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/5snv0yvvbes95",
  "venue": "arXiv",
  "versions": []
}