Erdős problem 1044

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 $\Lambda(f)$ .

Sources

The source's own material, retained exactly.

4 retained statements36c6d0fef71a

FormalConjectures/ErdosProblems/

1044.lean

Retained formal statement2 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 $f_n(z)=z^n-1$ .

FormalConjectures/ErdosProblems/1044.leanErdos1044.erdos_1044.variants.fixed_degree4 linesExact file

1∀ (n : ℕ),2  0 < n →3    IsLeast {L | ∃ f, Erdos1044.IsAdmissible f ∧ f.natDegree = n ∧ Erdos1044.maxBoundaryLength f = L}4      (Erdos1044.maxBoundaryLength (Polynomial.X ^ n - 1))

OpenStatement only, no proof

Source providers

Erdős Problems1
locator only
Formal Conjectures4
statement retained
VibeMathed0
not observed here
PLBY Lean proofs1
no statement retained
Jayyhk Erdős Lean1
no statement retained
William Blair Lean proofs0
not observed here
Erdős AI contributions wiki2
no statement retained
GPT-Erdős0
not observed here

Referenced paths

FormalConjectures/
- ErdosProblems/
  - 1044.lean4