Problem 1039

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claims/: The 5 claim pages of Problem 1039, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert \leq 1$ for all $i$. Let $\rho(f)$ be the radius of the largest disc which is contained in $\{z: \lvert f(z)\rvert< 1\}$.

Determine the behaviour of $\rho(f)$. In particular, is it always true that $\rho(f)\gg 1/n$?

Formulation. The site asks for the behavior of $\rho(f)$ and whether $\rho(f)\gg1/n$ always. Its source, Problem 3 of Erdős, Herzog and Piranian [EHP58] (p. 134), writes $\rho_n$ for the radius of the largest disk necessarily contained in $E$, that is $\rho_n=\inf\rho(f)$ over monic $f$ of degree $n$ with all zeros in the closed unit disk, asks for the asymptotic behavior of $\rho_n$ and whether $\rho_n>c/n$ for a positive constant $c$, and notes that $z^n-1$ shows $c\le\pi/2$. The page reads the first question as its source states it, the asymptotic behavior of the minimal inradius $\rho_n$, the reading of Krishnapur, Lundberg and Ramachandran [KLR25] as well; a claim that determines $\rho_n$ asymptotically answers the whole first question, and the second question is whether $\rho_n\gg1/n$. The source is carded at Metric properties of polynomials.

Status. Open (the site's label OPEN; page last edited 27 December 2025). The site's commentary credits Pommerenke [Po61] with $\rho(f)\ge1/(2en^2)$, the accepted partial claim on Pommerenke 1961, and Krishnapur, Lundberg and Ramachandran [KLR25] with $\rho(f)\gg1/(n\sqrt{\log n})$, the partial claim on Krishnapur, Lundberg and Ramachandran 2025. The site's proof-claims tab carries one full proof claim with no comments and no acceptance, on Geng–Qiu 2026: an arXiv preprint with a Lean companion, registered on the tab on 2026-09-09, asserts that $n\inf_{\deg f=n}\rho(f)\to\pi/2$. Two partial claims precede it: Price 2026, posted on the thread on 2026-05-07, claims $\rho(f)\ge(\log2)/n$, which would answer the second question in the affirmative; it was endorsed by readers on the thread and formalized by Kitamura, and is not marked accepted by the site. Houi 2026 claims $\rho(f)\ge1/(4n)$ when every critical value has modulus at least $1$. None of the 2026 claims is refereed or accepted by the site, and none has been built or audited in this wiki.

Source. erdosproblems.com/1039, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1039, https://www.erdosproblems.com/1039.

References.

• [EHP58] Erdős, P., Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. 6 (1958), 125--148; Problem 3, printed p. 134. Library home: erdos_1958_metric_properties_polynomials.
• [KLR25] M. Krishnapur, E. Lundberg, and K. Ramachandran, On the area of polynomial lemniscates. arXiv:2503.18270 (2025).
• [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; the paragraph recalling Problem 3 of Erdős, Herzog and Piranian, and Theorem 4, printed p. 101. Library home: pommerenke_1961_metric_properties_complex_polynomials (result page theorem_4).

Formalization. No statement in formal-conjectures; the community database listed none on 2026-10-06. Three Lean developments exist, each recorded at its pinned revision on its claim page: Houi's file, produced with Aristotle, which proves the non-degenerate case from axiomatized inputs; Kitamura's formalization of Price's argument; and the companion repository of the Geng–Qiu preprint. None has been built or audited in this wiki.

Current assessment

The question, as the site states it, asks for the behavior of the inradius $\rho(f)$ of $\{z:\lvert f(z)\rvert<1\}$ for monic $f$ of degree $n$ with all zeros in the closed unit disk, and in particular whether $\rho(f)\gg1/n$ always; Erdős, Herzog and Piranian noted that $z^n-1$ has $\rho\le(\pi/2)/n$, so the minimal inradius $\rho_n$ is at most $(\pi/2)/n$. The refereed record is Pommerenke's $\rho_n\ge1/(2en^2)$ [Po61], the accepted partial claim on Pommerenke 1961, and the preprint bound $\rho_n\gg1/(n\sqrt{\log n})$ of Krishnapur, Lundberg and Ramachandran [KLR25], the partial claim on Krishnapur, Lundberg and Ramachandran 2025, which the site credits without settling the problem. A thread claim of May 2026 offers to close the gap: Price's product argument, produced with GPT-5.5 Pro, argues that some zero is the center of a disk of radius $(\log2)/n$ inside the lemniscate, which would give $\inf_{\deg f=n}\rho(f)=\Theta(1/n)$; Sothanaphan digested it and vouched for it, the site's curator stated the inequality $\prod_j\lvert f(w_j)\rvert\le((1+\epsilon)^n-1)^n$ for points $w_j$ within $\epsilon$ of the zeros, from which the bound would follow, and Kitamura formalized the argument in Lean. The sharp constant is the pending full claim of Geng and Qiu (September 2026, AI-assisted, with a Lean companion): $n\inf\rho\to\pi/2$, so that $z^n-1$ would be asymptotically extremal. Read as its source states it (the Formulation above), the first question asks for the asymptotic behavior of the minimal inradius $\rho_n$, which Price's order and Geng and Qiu's constant together would settle; the earlier partial claim of Houi for polynomials without a critical point inside the lemniscate uses ideas that Lundberg says are already in the proof of Proposition 18(b) of [KLR25]. None of the 2026 claims is refereed or accepted by the site, whose page predates all of them; the postings' proofs are not compiled in this wiki. The derived standing is claimed/answered, through the pending full claim of Geng and Qiu; the accepted claims are partial.

Search scope: the site's problem page as exported (last edited 27 December 2025), its discussion thread (16 comments) and proof-claims tab (both read 2026-10-07), the community database entry (open), the arXiv record of the Geng–Qiu preprint, and the GitHub repositories linked from the thread and the tab; no formal-conjectures statement exists and no OpenAI release item names this problem. MathSciNet and zbMATH were not searched and X was not used.

Known Results

• Erdős, Herzog and Piranian: $f(z)=z^n-1$ has $\rho(f)\le(\pi/2)/n$, the upper obstruction the site records.
• [Po61], Theorem 4 (result page theorem_4): $\rho(f)\ge1/(2en^2)$; the accepted partial claim on Pommerenke 1961.
• [KLR25], Theorem 8: $\rho_n\gg1/(n\sqrt{\log n})$, the best bound the site's remarks cite; the partial claim on Krishnapur, Lundberg and Ramachandran 2025.
• Claimed, not accepted: Houi 2026, $\rho(f)\ge1/(4n)$ when every critical value has modulus at least $1$; Price 2026, $\rho(f)\ge(\log2)/n$ for every $f$, with the constant sharp for disks centered at zeros, hence $\inf_{\deg f=n}\rho(f)=\Theta(1/n)$; Geng–Qiu 2026, $n\inf_{\deg f=n}\rho(f)\to\pi/2$.

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.

• pommerenke_1961_metric_properties_complex_polynomials
• pommerenke_1961_metric_properties_complex_polynomials / theorem_4
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_3
• erdos_1958_metric_properties_polynomials / theorem_6
• krishnapur_2025_area_polynomial_lemniscates
• krishnapur_2025_area_polynomial_lemniscates / lemma_9
• krishnapur_2025_area_polynomial_lemniscates / theorem_8