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Erdős problem 1047

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Let f∈C[x]f\in \mathbb{C}[x] be a monic polynomial with mm distinct roots, and let c>0c>0 be a constant small enough such that {z:∣f(z)∣≤c}\{ z: \lvert f(z)\rvert\leq c\} has mm distinct connected components.

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FormalConjectures/ErdosProblems/

1047.lean

Retained formal statement5 of 6

The answer is no, as shown by Pommerenke [Po61], who showed that, if kk is sufficiently large, and f(z)=zk(z−a)f(z)=z^k(z-a) where aa is sufficiently close to (1+1k)k1k+1(1+\frac{1}{k})k^{\frac{1}{k+1}}, then {z:∣f(z)∣≤1}\{ z: \lvert f(z)\rvert\leq 1\} has two components, and the component which contains 00 is not convex.

FormalConjectures/ErdosProblems/1047.leanErdos1047.erdos_1047.variants.pommerenke12 linesExact file
∀ᶠ (k : ℕ) in Filter.atTop,  ∃ δ,    0 < δ ∧      ∀ (a : ℝ),        (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) < a →          a < (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) + δ →            (Erdos1047.componentsIn                    (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)).ncard =                2 ∧              ¬Convex ℝ                  (connectedComponentIn (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)                    0)
SolvedStatement only, no proof