Problem 511

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claims/: The 2 claim pages of Problem 511, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)\in \mathbb{C}[z]$ be a monic polynomial of degree $n$. Is it true that, for every $c>1$, the set

$$\{ z\in \mathbb{C} : \lvert f(z)\rvert< 1\}$$

has at most $O_c(1)$ many connected components of diameter $>c$ (where the implied constant is in particular independent of $n$)?

Status. Disproved, the site's label. The accepted claims are Pommerenke's Theorem 1 of 1961, refereed and credited by the site's curator, and Huang's independent 2025 construction, an arXiv preprint the curator credits as an independent proof.

Source. erdosproblems.com/511, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #511, https://www.erdosproblems.com/511.

References.

• [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
• [Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [Hu25] L. Huang, Many lemniscates with large diameter. arXiv:2509.11597 (2025).
• [Po28] G. Pólya, Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Phys.-math. Klasse (1928), 228-232 and 280-282.
• [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; Theorem 1, printed p. 98, stated there as the negative answer to Problems 8 and 9 of [EHP58]. Library home: pommerenke_1961_metric_properties_complex_polynomials; result page theorem_1.

Formalization. None recorded.

Current assessment

The answer is no. Pommerenke's Theorem 1 of 1961 [Po61] gives, for every $0<l<4$ and every $k$, a monic polynomial whose sublevel set has at least $k$ components of diameter at least $l$, and Huang's 2025 construction [Hu25] rediscovers the result independently by a different method. Both are accepted claims, Pommerenke 1961 on refereed publication and the site's credit and Huang 2025 on the site's credit alone, and either one refutes the question for every $1<c<4$: the components of diameter above $c$ cannot be bounded independently of the degree. The transfer from the closed sublevel sets of the sources to the site's open set is recorded on the claim pages.

Search and coverage. The standing rests on the site page and its commentary as accessed and on the two papers [Po61] and [Hu25]; no wider literature search is recorded. The claim pages state both theorems as the papers print them; neither proof has been independently reviewed by this corpus, and the rescaling step that Pommerenke's proof leaves unstated, where the approximation theorem quoted at capacity one is applied to a set of smaller capacity, is not reconstructed in this corpus. The question has content only for $1<c<4$, by Pólya's bound [Po28], and both constructions cover that whole range, so no part of the question remains open.

Known Results

• Pólya [Po28], as both constructions cite it, bounds the diameter of every component of the sublevel set of a monic polynomial by $4$, so the range $c<4$ of the constructions cannot be enlarged and the question has content only for $1<c<4$.
• Pommerenke [Po61], Theorem 1: for every $0<l<4$ and every $k\ge1$ a monic polynomial whose sublevel set has at least $k$ components of diameter at least $l$, stated as the negative answer to Problems 8 and 9 of [EHP58]. This is the accepted claim Pommerenke 1961.
• Huang [Hu25], Theorem 1.1: for every $c\in(0,4)$ and every $N$ a monic polynomial whose sublevel set has at least $N$ components of diameter at least $c$, by the Hilbert lemniscate theorem applied inside a domain of logarithmic capacity one. This is the accepted claim Huang 2025.

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.

• huang_2025_many_lemniscates_large_diameter
• huang_2025_many_lemniscates_large_diameter / theorem_1_1
• pommerenke_1961_metric_properties_complex_polynomials
• pommerenke_1961_metric_properties_complex_polynomials / theorem_1
• erdos_1961_unsolved_problems
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_9
• erdos_1958_metric_properties_polynomials / problem_p148