Problem 1044

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claims/: The 1 claim page of Problem 1044, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $\Lambda(f)$.

Status. 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). 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.

Source. erdosproblems.com/1044, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1044, https://www.erdosproblems.com/1044.

References.

• [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
• [Ta26] Tang, Q., On Erdős Problem #1044. Note with TeX source, https://github.com/QuanyuTang/erdos-problem-1044, uploaded 2026-01-05; unrefereed. Cited on the claim page above at its pinned address.

Formalization. Statement in formal-conjectures, read at its revision of 2026-09-18: it credits Tang, tags the problem and its infimum variant as solved with their proofs left as sorry, and its formal_proof attribute on both points to the copy of Luccioli's development in Alexeev's repository, linked from the claim page above together with the gist; the fixed-degree variant is tagged open.

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_11