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 statement4 of 8

If $1/2<r\leq \frac{\sqrt{5}-1}{2}$ then the component which contains $0$ must have diameter $>1/r$ .

FormalConjectures/ErdosProblems/1048.leanErdos1048.erdos_1048.variants.diam_gt_inv_r8 linesExact file

1∀ (r : ℝ),2  1 / 2 < r →3    r ≤ (√5 - 1) / 2 →4      ∀ (f : Polynomial ℂ),5        f.Monic →6          f.degree ≥ 1 →7            (∀ z ∈ f.roots, ‖z‖ ≤ r) →8              ENNReal.ofReal (1 / r) < Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) 0)

SolvedStatement only, no proof

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