Erdős problem 509 · problems.science

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. Can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of closed discs the sum of whose radii is $≤ 2$ ?

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FormalConjectures/ErdosProblems/

509.lean

Retained formal statement5 of 5

Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial. If it is connected, can the set $\{z ∈ ℂ : |f(z)| ≤ 1\}$ be covered by a set of circles the sum of whose radii is $≤ 2$ ? Solution: True. This is due to Pommerenke.

FormalConjectures/ErdosProblems/509.leanErdos509.erdos_509.variants.Pommerenke_connected6 linesExact file

1True ↔2  ∀ (f : Polynomial ℂ),3    f.Monic →4      f.natDegree ≠ 0 →5        IsConnected {z | ‖Polynomial.eval z f‖ ≤ 1} →6          ∃ ι, Nonempty (Erdos509.BoundedDiscCover {z | ‖Polynomial.eval z f‖ ≤ 1} 2 ι)

SolvedStatement only, no proof

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