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PreprintCC BY 4.0arXiv

Pierce-Birkhoff conjecture is false

  • Zehua Lai
  • Lek-Heng Lim
  • Junyu Ren
Open at sourcePDFDOICite

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 ℝn 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.

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Cite

BibTeXRIS
BibTeX
@misc{lai2026piercebirkhoff,
  title = {{Pierce-Birkhoff conjecture is false}},
  author = {Zehua Lai and Lek-Heng Lim and Junyu Ren},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10420},
  eprint = {2609.10420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10420},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Pierce-Birkhoff conjecture is false
AU  - Zehua Lai
AU  - Lek-Heng Lim
AU  - Junyu Ren
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10420
UR  - https://arxiv.org/abs/2609.10420
AB  - 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 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.
ER  -