{
  "abstract": "We provide a counterexample to the Pierce-Birkhoff conjecture: a continuous function that is piecewise quadratic on a finite collection of semialgebraic sets that partition $\\mathbb{R}^n$ but that cannot be expressed as a finite lattice combination of polynomials. The counterexample was found with the assistance of our multi-agent, multi-model setup that chains together GPT 5.6, GPT 6, Claude Opus 5, and Claude Fable 5.1.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.10420",
  "author_total": 3,
  "authors": [
    {
      "name": "Zehua Lai"
    },
    {
      "name": "Lek-Heng Lim"
    },
    {
      "name": "Junyu Ren"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9g2q34czkan71.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.10420",
  "fulltext_url": null,
  "id": "9g2q34czkan71",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.10420",
  "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.AG"
    ],
    [
      "openalex",
      "1703|Polynomial and algebraic computation"
    ]
  ],
  "openalex": "W7212186197",
  "pdf_url": "https://arxiv.org/pdf/2609.10420",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9g2q34czkan71.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:16Z",
      "native_id": "2609.10420",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.10420"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Pierce-Birkhoff conjecture is false",
  "title_text": "Pierce-Birkhoff conjecture is false",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:16Z",
  "url": "https://research.draconitic.com/paper/9g2q34czkan71",
  "venue": "arXiv",
  "versions": []
}