Pierce–Birkhoff conjecture

formal-conjectures · wikipedia · ams-13

The Pierce-Birkhoff conjecture states that for every real piecewise-polynomial function
f : ℝⁿ → ℝ, there exists a finite set of polynomials gᵢⱼ ∈ ℝ[x₁, ..., xₙ] such that
f = supᵢ infⱼ(gᵢⱼ).

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Pierce%E2%80%93Birkhoff_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/PierceBirkhoff.lean
Local target: Corpus.WikipediaPierceBirkhoff.pierce_birkhoff_conjecture
Source SHA-256: 4ec19158a4d0ce748d836d62494a8f26ef446f58150c1f0b85b7d6f4f151cf98
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record