Erdős Problem 1150

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$ , $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

Field theory and polynomials AMS 12 AMS 30

Mathematical statement

Is there some constant $c > 0$ such that, for all large enough $n$ and all polynomials $P$ of degree $n$ with coefficients in $\{-1, 1\}$ , $\max_{|z|=1} |P(z)| > (1 + c) \sqrt{n}?$

Statement source: Erdős Problems statement material

Statement terms: Source-specific

Source-specific terms. Therefore does not assert reuse rights beyond attributed display.

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

erdos_1150

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_1150 :    answer(sorry) ↔ ∃ c > 0, ∀ᶠ n in Filter.atTop,      ∀ P : ℂ[X],  (∀ i ≤ P.natDegree, P.coeff i = - 1 ∨ P.coeff i = 1) → P.natDegree = n →        ⨆ z : Metric.sphere (0 : ℂ) 1, ‖P.eval (z : ℂ)‖ > (1 + c) * Real.sqrt n := 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

Pinned Formal Conjectures sourceformalization erdosproblems.com/1150background