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 statement3 of 5

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 $≤ 2e$ ? Solution: True. This is due to Cartan. See *Sur les systèmes de fonctions holomorphes à variétés linéaires lacunaires et leurs applications*, Henri Cartan, http://www.numdam.org/article/ASENS_1928_3_45__255_0.pdf

FormalConjectures/ErdosProblems/509.leanErdos509.erdos_509.variants.Cartan_bound4 linesExact file

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

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