Problem 114

If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$ ?

Source: erdosproblems.com/114

Polynomials
Analysis

Wiki page

Status

Open

No claim settles this problem.

Falsifiable. The site's label is FALSIFIABLE (page last edited 2026-01-23): open, but refutable by a finite counterexample. Six partial claims are recorded, none settling the question. Degree $2$ is proved twice, by Wang (Wang 1998, accepted, refereed) and by Eremenko and Hayman (Eremenko–Hayman 1999, accepted, refereed). All sufficiently large degrees are proved by Tao in an arXiv preprint (Tao 2025, pending). Degree $3$ has two pending claims: Dahlke's Zenodo manuscript, announced on the site's discussion thread on 2026-05-21 (Dahlke 2026), and Chatelet's write-up and Lean development, posted in August 2026 and registered on the site's proof-claims tab on 2026-09-07 (Chatelet 2026), which asserts the cubic case under a domain reduction cited from Eremenko–Hayman and Tao; no review or acceptance of either is recorded. Degrees $3$ to $14$ were claimed by Mendoza's computer search, withdrawn by its author on 2026-10-01 (Mendoza 2026).

Bo95: Borwein, Peter, The arc length of the lemniscate $\{|p(z)|=1\}$ . Proc. Amer. Math. Soc. (1995), 797-799.
Da07: Danchenko, V. I., The lengths of lemniscates. Variations of rational functions. Mat. Sb. 198 (2007), no. 8, 51-58.
Do61: Dol\v zenko, E. P., Some estimates concerning algebraic hypersurfaces and derivatives of rational functions. Dokl. Akad. Nauk SSSR (1961), 1287-1290.
EHP58: Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
ErHa99: Eremenko, Alexandre and Hayman, Walter, On the length of lemniscates. Michigan Math. J. 46 (1999), no. 2, 409-415.
FrNa09: Fryntov, Alexander and Nazarov, Fedor, New estimates for the length of the Erd\H os-Herzog-Piranian lemniscate. In Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, 226 (2009), 49-60.
Ha74: Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
Po59: Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227. Theorem 2, p. 222: "If $E$ is connected, the length of $C$ is at least $2\pi$ , with equality only for $f(z)=z^n$ ", $C$ the lemniscate $|f(z)|=1$ and $E$ its interior; this is the result behind the site's remark that Pommerenke [Po59] settled the connected-case question of [EHP58]. The paper does not treat this page's question, whether $z^n-1$ maximizes the length. Library home: Pommerenke 1959 and its Pommerenke 1959, Theorem 2 page.
Po61: Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; the sentence on Problem 12a and Theorem 9, p. 104. Library home: Pommerenke 1961 and its result page Pommerenke 1961, Theorem 9.
Ta25: T. Tao, The maximal length of the Erdős-Herzog-Piranian leminscate length in high degree. arXiv:2512.12455 (2025).
Va99: Various, Some of Paul's favorite problems. Booklet produced for the conference "Paul Erdős and his mathematics", Budapest, July 1999 (1999).
Wa98: Wang, Chunjie, The arc length of the lemniscate $\lvert w^2+c\rvert=1$ . Acta Math. Sci. (Chin. Ed.) 18 (1998), no. 3, 297–301; Zbl 0926.31001.

Problem 114

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