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arXiv · 2609.10420

Pierce-Birkhoff conjecture is false

Zehua Lai·Lek-Heng Lim·Junyu Ren
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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.

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Zehua Lai, Lek-Heng Lim, Junyu Ren. 2026-09-09. Pierce-Birkhoff conjecture is false. https://arxiv.org/abs/2609.10420

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