Erdős problem 1044 · problems.science

Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$ . If $\Lambda(f)$ is the maximum of the lengths of the boundaries of the connected components of $\{ z: \lvert f(z)\rvert<1\}$ then determine the infimum of $\Lambda(f)$ .

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4 retained statements2415f78e850a

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FormalConjectures/ErdosProblems/

1044.lean

Retained formal statement3 of 4

Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$ (and proves this for $n=1$ and $n=2$ ).

FormalConjectures/ErdosProblems/1044.leanErdos1044.erdos_1044.variants.fixed_degree_of_le_two4 linesExact file

1∀ (n : ℕ),2  n = 1 ∨ n = 2 →3    IsLeast {L | ∃ f, Erdos1044.IsAdmissible f ∧ f.natDegree = n ∧ Erdos1044.maxBoundaryLength f = L}4      (Erdos1044.maxBoundaryLength (Polynomial.X ^ n - 1))

SolvedStatement only, no proof

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