Erdős problem 1047 · problems.science

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

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6 retained statements2415f78e850a

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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 $k$ is sufficiently large, and $f(z)=z^k(z-a)$ where $a$ is sufficiently close to $(1+\frac{1}{k})k^{\frac{1}{k+1}}$ , then $\{ z: \lvert f(z)\rvert\leq 1\}$ has two components, and the component which contains $0$ is not convex.

FormalConjectures/ErdosProblems/1047.leanErdos1047.erdos_1047.variants.pommerenke12 linesExact file

1∀ᶠ (k : ℕ) in Filter.atTop,2  ∃ δ,3    0 < δ ∧4      ∀ (a : ℝ),5        (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) < a →6          a < (1 + 1 / ↑k) * ↑k ^ (1 / (↑k + 1)) + δ →7            (Erdos1047.componentsIn8                    (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)).ncard =9                2 ∧10              ¬Convex ℝ11                  (connectedComponentIn (Erdos1047.sublevelSet (Polynomial.X ^ k * (Polynomial.X - Polynomial.C ↑a)) 1)12                    0)

SolvedStatement only, no proof

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