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 $\Lambda(f)$ .

Source: erdosproblems.com/1044

Analysis

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SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

SOLVED (LEAN) on the site. Tang's note of 2026-01-05 proves that the infimum of $\Lambda(f)$ is $2$ ; the site's curator credits it and Tao read the argument, and it is the accepted claim (Tang's claim page (2026)). The Lean qualifier of the site's label is Luccioli's formalization of Tang's note with Aristotle, third-party Lean not built here. The fixed-degree question Tang raises, whether $z^n-1$ minimizes $\Lambda$ in each degree $n$ , is settled only for $n\le2$ and is not part of the problem.

Problem 1044

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