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 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 $f$ .

FormalConjectures/ErdosProblems/1047.leanErdos1047.erdos_1047.variants.max_non_convex_components10 linesExact file

1∀ (n : ℕ),2  IsGreatest3    {k |4      ∃ f c,5        f.Monic ∧6          f.natDegree = n ∧7            0 < c ∧8              (Erdos1047.componentsIn (Erdos1047.sublevelSet f c)).ncard = (f.rootSet ℂ).ncard ∧9                {t | t ∈ Erdos1047.componentsIn (Erdos1047.sublevelSet f c) ∧ ¬Convex ℝ t}.ncard = k}10    sorry

OpenStatement only, no proof

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