Maximal length of a lemniscate: is $z^n-1$ the extremal monic polynomial of degree $n$? (Erdős #114)
Statement
Let $p(z)\in\mathbb{C}[z]$ be a monic polynomial of degree $n$, and consider its lemniscate $$\Lambda(p)=\{z\in\mathbb{C} : \lvert p(z)\rvert = 1\},$$ a level curve of $\lvert p\rvert$ whose total arc length (one-dimensional Hausdorff measure) is finite. Is the length of $\Lambda(p)$, among all monic polynomials of degree $n$, maximized by $p(z)=z^n-1$? Writing $f(n)$ for the supremum of the lemniscate length over all monic degree-$n$ polynomials, the conjecture asserts that $f(n)$ is attained at $z^n-1$ (up to the trivial symmetries of rotating and translating the variable). A single monic polynomial whose lemniscate is certifiably longer than that of $z^n-1$ of the same degree would disprove it.
Acceptance. FULLY RESOLVES: a proof that for every $n$, no monic degree-$n$ polynomial has lemniscate length exceeding that of $z^n-1$ — machine-checkable (Lean/Coq) preferred, else a complete written proof; a hybrid resolution (an effective version of the large-$n$ uniqueness theorem plus a rigorous, certified computation covering the remaining finite range of degrees) is acceptable if every step is rigorous and reproducible. OR a counterexample: an explicit monic polynomial $p$ of stated degree $n$ (exact coefficients) with a certified computation — interval arithmetic or exact analysis, not bare floating point — showing its lemniscate length strictly exceeds the certified length of the lemniscate of $z^n-1$. ADVANCES: (a) a rigorous proof of the conjecture for specific new degrees beyond the known $n=2$ case (e.g., $n=3$ or $n=4$), with reproducible certification code; (b) an explicit all-$n$ upper bound on $f(n)$ strictly better than the best all-$n$ bound stated in the background, with proof; (c) an effective explicit threshold $n_0$ for the large-$n$ uniqueness result; (d) certified high-precision values of $f(n)$ for small $n$ with rigorous enclosures. Deliver the proof file, or the counterexample polynomial plus certification code, or the per-degree certificates and code with attained bounds.
Background
The lemniscate length problem of Erdős, Herzog, and Piranian [EHP58, p.142], repeated by Erdős across four decades [Er61, p.247; Er82e; Er90; Er97f], listed as Problem 4.10 in Hayman's problem collection [Ha74] and as Problem 2.35 in [Va99]; Erdős offered $250 for a solution (prize reported by Borwein [Bo95]). Listed as open on erdosproblems.com/114 (fetched 2026-07-13, status 'falsifiable'). The known frontier is long and now nearly closes the problem: the length for $z^n-1$ is $2n+O(1)$, so the conjecture implies $f(n)=2n+O(1)$. Dolzhenko [Do61] proved $f(n)\le 4\pi n$ (a result long overlooked); Pommerenke [Po61] proved $f(n)\ll n^2$; Borwein [Bo95] independently recovered $f(n)\ll n$; Eremenko and Hayman [ErHa99] proved the full conjecture for $n=2$ and the explicit bound $f(n)\le 9.173\,n$ for all $n$; Danchenko [Da07] improved the all-$n$ bound to $f(n)\le 2\pi n$; Fryntov and Nazarov [FrNa09] proved that $z^n-1$ is a local maximizer and settled the problem asymptotically with $f(n)\le 2n+O(n^{7/8})$. Most recently Tao [Ta25] proved that $z^n-1$ is the unique maximizer (up to rotation and translation) for all sufficiently large $n$ — so the surviving open content is the conjecture for every fixed $n\ge 3$ (only $n=2$ is settled unconditionally) and an effective version of Tao's threshold. A companion EHP question — the length is at least $2\pi$ whenever $\{\lvert p\rvert<1\}$ is connected — was proved by Pommerenke [Po59]. This problem is a sibling of the venue problem on short root-connecting paths inside $\{\lvert f\rvert<1\}$ (Erdős #1041), from the same [EHP58] paper. The attacker's tool: certified global optimization for fixed small degree — normalize the coefficient space, branch-and-bound with interval arithmetic, and certify lemniscate arc length by rigorous quadrature — to settle $n=3,4,\dots$; or make Tao's large-$n$ threshold effective and close the finite gap by rigorous computation.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #114 (T. F. Bloom) | website |
| REF-02 | T. Tao, The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree (arXiv:2512.12455) [Ta25] | arxiv |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.