Pierce–Birkhoff conjecture for piecewise polynomial functions
Can every continuous, piecewise polynomial function defined on a finite semialgebraic partition of Euclidean space be expressed as a finite lattice combination of polynomials using addition, subtraction, multiplication, and maximum/minimum?
Model / AI: GPT-5.6, GPT-6, Claude Opus 5, Claude Fable 5.1
•self-reported (counterexample)
Sep 9, 2026
Piecewise quadratic counterexample
Model / AI: GPT-5.6, GPT-6, Claude Opus 5, Claude Fable 5.1
•self-reported (counterexample)
Zehua Lai, Lek-Heng Lim, and Junyu Ren construct a counterexample disproving the 1956 Pierce–Birkhoff conjecture. The authors report employing a multi-agent, multi-model workflow where models assisted in exploring candidate piecewise quadratic configurations and verifying non-expressibility as polynomial lattice combinations.
Source read 2026-10-08. Read the arXiv v1 abstract, construction outline, and AI-use disclosure. The full algebraic calculations were not independently reproduced.