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Claim. There is $n_0$ such that for every $n\ge n_0$ and every monic polynomial $p$ of degree $n$ the length of the lemniscate $\{z:\lvert p(z)\rvert=1\}$ is at most that of $p_0(z)=z^n-1$, with equality exactly when $p(z)=(z-z_0)^n-e^{i\theta}$, that is, when $p$ is $p_0$ up to rotation and translation. This is part (iv) of Theorem 1.1 of Tao, The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree, arXiv:2512.12455 (v1 2025-12-13, v2 2025-12-22), and it settles the question of Problem 114 for all large degrees. Parts (i) to (iii) give the chain of bounds $2n+O(\sqrt n)$, $2n+O(1)$ and $2n+4\log2+o(1)$ for the maximal length, the last matching the length $2^{1/n}B(\tfrac12,\tfrac1{2n})=2n+4\log2+O(1/n)$ of the lemniscate of $p_0$; the argument builds on the analysis of Fryntov and Nazarov, who had $2n+O(n^{7/8})$, and its Remark 1.2 describes the lemniscate of $p_0$ as $2n$ spokes of length about one with $n$ tips resembling semicircles of radius $1/n$, a shape that Heuristic 1.1 expects near-extremal polynomials to approach. Remark 1.3 says that every implied constant is effectively computable, so the full conjecture reduces to checking an explicitly bounded number of degrees, and that only one scenario could keep that check from finishing in finite time: a competitor of bounded degree whose lemniscate has exactly the length of that of $p_0$. The digest is on the source card tao_2025_maximal_length_erdos_herzog_piranian_lemniscate.

Covers. Every degree above an effectively computable bound $n_0$, which the paper does not compute. Degree $2$ is settled by Eremenko and Hayman (Eremenko–Hayman 1999); the degrees from $3$ to $n_0$ are not covered, so the claim settles the conjecture for all but finitely many degrees without deciding it.

Depends on. No page of this wiki.

Acceptance. None recorded. The paper is an arXiv preprint with no journal version on its card, so no refereed evidence exists. The site's commentary (page last edited 2026-01-23) records that Tao proved $z^n-1$ to be the unique maximizer up to rotation and translation for all sufficiently large $n$, but its label is FALSIFIABLE, an open label, so the remark is commentary and not acceptance and no reviewed evidence is listed. No formal proof is recorded. Nothing here is this project's own review.