Problem 511

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 $n$ )?

Source: erdosproblems.com/511

Analysis

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Status

SolvedDisproved

An accepted solution exists. The statement is false.

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.

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; result page Pommerenke 1961, Theorem 1.

Problem 511

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