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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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6 retained statements36c6d0fef71a

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FormalConjectures/ErdosProblems/

1047.lean

Retained formal statement4 of 6

Goodman raises the question of the maximum number of non-convex components that are possible as a function of the degree of ff.

FormalConjectures/ErdosProblems/1047.leanErdos1047.erdos_1047.variants.max_non_convex_components10 linesExact file
∀ (n : ℕ),  IsGreatest    {k |      ∃ f c,        f.Monic ∧          f.natDegree = n ∧            0 < c ∧              (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = (f.rootSet ℂ).ncard ∧                {t | t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c) ∧ ¬Convex ℝ t}.ncard = k}    sorry
OpenStatement only, no proof