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 statement6 of 8

Pommerenke [Po61] proved the answer is no for $r>1$ , showing that if $f(z)=z^n-r^n$ then $\{ z: \lvert f(z)\rvert \leq 1\}$ has $n$ connected components, all with diameter $\to 0$ as $n\to \infty$ .

FormalConjectures/ErdosProblems/1048.leanErdos1048.erdos_1048.variants.pommerenke_diam_tendsto_zero9 linesExact file

1∀ (r : ℝ),2  1 < r →3    ∀ (ε : ℝ),4      0 < ε →5        ∀ᶠ (n : ℕ) in Filter.atTop,6          ∀ (f : Polynomial ℂ),7            f = Polynomial.X ^ n - Polynomial.C (↑r ^ n) →8              ∀ z ∈ Erdos1043.levelSet f,9                Metric.ediam (connectedComponentIn (Erdos1043.levelSet f) z) < ENNReal.ofReal ε

SolvedStatement only, no proof

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