Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 114

claims/: The 6 claim pages of Problem 114, one per claimant's result; the problem's standing derives from them.

Statement. 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 $p(z)=z^n-1$ ?

Status. 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 Degrees $3$ to $14$ were claimed by Mendoza's computer search, withdrawn by its author on 2026-10-01 (Mendoza 2026).

Source. erdosproblems.com/114, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #114, https://www.erdosproblems.com/114.

References.

[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_some_problems_erdos_herzog_piranian and its 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_metric_properties_complex_polynomials and its result page 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.

Formalization. None recorded: formal-conjectures holds no statement file for the problem.

Current assessment

The site labels the problem FALSIFIABLE: a single monic polynomial whose lemniscate is longer than that of $z^n-1$ of the same degree would refute the conjecture by a finite computation of two lengths. Remark 1.3 of Tao 2025 adds that the constants of his proof are effective, so the conjecture reduces to finitely many degrees, and that only an exact tie between the lemniscate lengths of $z^n-1$ and of a competitor in some bounded degree could keep it from being decided by a finite computation. The site's remarks record the question as settled for $n=2$ by Eremenko and Hayman [ErHa99], a case Wang [Wa98] had proved a year earlier in a paper a comment on the site's discussion thread pointed to, and for all sufficiently large $n$ by Tao [Ta25], with $z^n-1$ the unique maximizer up to rotation and translation; each result settles the instances it names and has its own partial claim page, the first two accepted on their refereeing (Wang 1998, Eremenko–Hayman 1999), the third pending as an arXiv preprint (Tao 2025); because the site's label is an open one, its remarks are commentary and not acceptance. The general upper bounds $f(n)\le 2\pi n$ [Da07] and $f(n)\le 2n+O(n^{7/8})$ [FrNa09] precede Tao's result. The degrees from $3$ up to Tao's bound remain open; two pending claims treat degree 3, Dahlke's manuscript (Dahlke 2026) and Chatelet's write-up (Chatelet 2026), the latter's Lean development at the pinned commit stating its cited reduction as a hypothesis that is stronger than the cited results and false, so that its formal theorem holds vacuously (the claim page records the details). Mendoza's interval-arithmetic search over degrees $3$ to $14$ (Mendoza 2026) was withdrawn by its author on 2026-10-01, after an objection on its code repository showed that its lengths and margins were estimates rather than bounds and that half of the coefficient box was never evaluated for $n\ge6$ . A research note on degree $4$ posted on the thread on 2026-08-10 says itself that it does not solve that case; it settles no degree and has no claim page. Search scope: the site's references, its proof-claims tab and its discussion thread (nine comments, the last of 2026-09-05), the Zenodo records of Dahlke and Mendoza, the Mendoza repository and its issue #4, Chatelet's Zenodo record and repository, and the zbMATH record of Wang 1998; no wider literature search is recorded. Proof coverage: none of the proofs is reconstructed or compiled in this corpus.

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.

Linked from (36)

The lemniscate conjecture in degree 2, by Wang The lemniscate conjecture in degree 2 The lemniscate conjecture in all sufficiently large degrees Mendoza's computer search over degrees 3 to 14, withdrawn The lemniscate inequality for cubics The lemniscate conjecture for cubics, under a cited domain reduction problems/polynomials/E0114/claims Polynomials analysis/pommerenke_1959_some_problems_erdos_herzog_piranian Theorem 2: a connected lemniscate interior gives length at least 2π, with equality only for z^n analysis/pommerenke_1961_metric_properties_complex_polynomials Theorem 9: the lemniscate |f(z)| = 1 has length less than 74 n^2 Polynomials polynomials/danchenko_2007_lengths_lemniscates Lemma 1 (p. 53): length is at most secant variation times analytic capacity Theorem 1 (p. 52): every lemniscate of a monic degree-n polynomial at level r^n has length at most 2 pi n r Theorem 2 (pp. 56-57): the variation of a rational function of degree at most n on a curve is at most n Psi times its sup norm polynomials/erdos_1958_metric_properties_polynomials Problem 12: is the lemniscate |f| = 1 longest for z^n − 1, at least 2π when E is connected, and how short for zeros in D?polynomials/eremenko_1999_length_lemniscates Lemma 1 (p. 2): the preimage of a circle under a degree d rational map meets a circle at most 2d times Lemma 4 (p. 3): the lemniscate length is continuous in the coefficients and has a maximizer in each degree Lemma 5 (p. 5): some extremal polynomial has all its critical points on its lemniscate Lemma 6 (p. 7): some extremal polynomial has a connected lemniscate Remark (p. 5): z^2 + 1 is extremal for d = 2, its level set the Bernoulli lemniscate Theorem 1 (p. 1): the lemniscate of a monic degree d polynomial has length at most alpha_0 d < 9.173 d Theorem 2 (p. 2): the preimage of a circle under a degree d rational map has spherical length at most d great circles polynomials/fryntov_2009_new_estimates_length_erdos_herzog Improved upper bound (Section 8, pp. 10--11): the lemniscate of a monic degree-n polynomial has length at most 2 pi (n - 1 + sqrt n)Asymptotic estimate (Section 9, pp. 11--17): the lemniscate of a monic degree-n polynomial has length at most 2n + O(n^{7/8})Local maximality (stated p. 4, proved in Section 6, pp. 7--10): z^n - 1 locally maximizes the lemniscate length among monic polynomials polynomials/tao_2025_maximal_length_erdos_herzog_piranian_lemniscate Lemma 3.2 (p. 20): the length of the lemniscate of z^n - 1 as a beta integral, and its length inside a small disk Proposition 1.2 (p. 5): a maximizer of lemniscate length exists, with connected lemniscate through all critical points, and may be normalized Theorem 1.1 (p. 4): the lemniscate of a monic degree n polynomial has length at most 2n + 4 log 2 + o(1), and for large n at most that of z^n - 1 polynomials/wang_2026_proposed_complete_solution_erdos_problem_1038

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