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Claim. For every monic polynomial $p$ of degree $2$ the length of the lemniscate $E(p)=\{z:\lvert p(z)\rvert=1\}$ is at most that of $z^2+1$, whose lemniscate is the Bernoulli lemniscate; since $z^2+1$ is $z^2-1$ up to rotation, this is the degree-2 case of the question of Problem 114. Eremenko and Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), no. 2, 409–415, state it in the abstract and derive it in the remarks after their Lemma 5, which gives an extremal polynomial all of whose critical points lie on its lemniscate (Lemma 4 shows that a maximum of the length exists): a monic quadratic has one critical point, so an extremal quadratic has critical value of modulus one and is $z^2+1$ up to rotation and translation, whose lemniscate has length $2^{3/2}\int_{-1}^{1}(1-x^4)^{-1/2}\,dx\approx7.416$. The same paper proves the general bound $\lvert E(p)\rvert\le\alpha_0 d<9.173\,d$ for monic $p$ of degree $d$ (Theorem 1), where $\alpha_0$ is the supremum of the perimeters of convex hulls of compact connected sets of logarithmic capacity $1$. The digest is on the source card eremenko_1999_length_lemniscates; the page's date is the issue's month as Crossref records it, September 1999, and the issue gives no day, so the first of the month stands in for it.

Covers. Degree $2$ only: $z^2-1$ maximizes the lemniscate length among monic quadratics. The paper's Theorem 1 bounds every degree but settles no other instance of the conjecture; the case of all sufficiently large degrees is Tao's (Tao 2025), and the intermediate degrees are open.

Depends on. No page of this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in the Michigan Mathematical Journal, volume 46, issue 2 (1999). The site's commentary records that Eremenko and Hayman proved the full conjecture for $n=2$, 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, so no formalized evidence is listed. Nothing here is this project's own review.