Erdős problem 1048 · problems.science

If $f\in \mathbb{C}[x]$ is a monic polynomial with all roots satisfying $\lvert z\rvert \leq r$ for some $r<2$ , then must $\{ z: \lvert f(z)\rvert <1\}$ have a connected component with diameter $>2-r$ ?

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8 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

1048.lean

Retained formal statement2 of 8

If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$ , which $f(z)=z^n$ shows is best possible.

FormalConjectures/ErdosProblems/1048.leanErdos1048.erdos_1048.variants.diam_ge_two6 linesExact file

1∀ (r : ℝ),2  0 ≤ r →3    r ≤ 1 / 2 →4      ∀ (f : Polynomial ℂ),5        f.Monic →6          f.degree ≥ 1 → (∀ z ∈ f.roots, ‖z‖ ≤ r) → 2 ≤ Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)

SolvedStatement only, no proof

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