Pierce–Birkhoff conjecture
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 statement
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).
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Statement artifacts, not proofs
These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.
Pinned Lean formulation 1
pierce_birkhoff_conjecture
theorem pierce_birkhoff_conjecture {n : ℕ} (f : (Fin n → ℝ) → ℝ) (hf : IsPiecewiseMvPolynomial f) : ∃ (ι κ : Type) (g : ι → κ → MvPolynomial (Fin n) ℝ), Finite ι ∧ Finite κ ∧ ∀ x, f x = ⨆ i, ⨅ j, MvPolynomial.eval x (g i j) := by sorry- Statement source
- Formal Conjectures
- Lean version
- v4.27.0
- Placeholder
- Present; no proof artifact
- Source evidence
- Pinned source index
- Fidelity review
- Community formulation
References