Erdős Problem 1041

Let $f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x]$ with $|z_i| < 1$ for all $i$ .

Mathematical statement

Let $f(z) = \prod_{i=1}^{n} (z - z_i) \in \mathbb{C}[x]$ with $|z_i| < 1$ for all $i$ .

Conjecture: Must there always exist a path of length less than 2 in $\{ z \in \mathbb{C} \mid |f(z)| < 1 \}$ which connects two of the roots of $f$ ?

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_1041

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem erdos_1041 :    ∃ (z₁ z₂ : ℂ) (h : ({z₁, z₂} : Multiset ℂ) ≤ f.roots) (γ : Path z₁ z₂),      Set.range γ ⊆ { z : ℂ | ‖f.eval z‖ < 1 } ∧ length (Set.range γ) < 2 := 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/1041background