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Statement

Normalization (Remark 1.1, p. 2). The length of the lemniscate $\partial E_1(p)=\{z:\lvert p(z)\rvert=1\}$ does not change when $p(z)$ is replaced by a translate $p(z-z_0)$ or a rotation $e^{-in\theta}p(e^{i\theta}z)$, $z_0\in\mathbb C$, $\theta\in\mathbb R$. A polynomial is normalized when its $z^{n-1}$ coefficient vanishes and its constant coefficient is a non-positive real.

Proposition 1.2 (p. 5, quoted). "Let $n\geq1$. Then there exists a monic polynomial $p$ of degree $n$ which maximizes $\ell(\partial E_1(p))$ among all such polynomials. Furthermore, the lemniscate $\partial E_1(p)$ is connected and contains all the critical points of $p$. Finally, we can assume $p$ to be normalized in the sense of Remark 1.1."

The proposition asserts these properties for some maximizer, not for every maximizer. The paper calls a polynomial with all the properties of the proposition a normalized maximizer (p. 5), observes that for $n=2$ the only one is $z^2-1$, which is why the case $n=2$ of the conjecture follows from Eremenko and Hayman's work, and notes, with an example in Figure 3 (p. 6), that for $n\ge3$ these properties do not determine the normalized maximizer.

Proof pointer

P. 5. The existence of a maximizer whose lemniscate is connected and contains all its critical points is taken from Lemmas 5 and 6 of Eremenko and Hayman; the paper's footnote 1 says their proofs use quasiconformal mappings and the Riemann–Hurwitz formula. A translation then removes the $z^{n-1}$ coefficient and a rotation makes $p(0)$ a non-positive real. The same footnote says that containing the critical points is not essential to the paper's arguments (the Gauss–Lucas theorem can substitute), while connectedness is used in Section 4 and in the proof of Lemma 11.1.

Read depth

Claims checked: Remark 1.1, Proposition 1.2, its proof and footnote 1 were read clause by clause on the page images of arXiv:2512.12455v2. Lemmas 5 and 6 of Eremenko and Hayman were not read. Nothing here is independently reviewed.

Dependencies

External: A. Eremenko and W. Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), 409--415, Lemmas 5 and 6.

Source. Terence Tao, The maximal length of the Erdős–Herzog–Piranian lemniscate in high degree, arXiv:2512.12455 (2025), version v2 of 22 December 2025; the edition read is named on the source card.

Bears on

• Problem 114: reduces the problem in each degree to normalized maximizers, whose lemniscates are connected and pass through every critical point; it is the reduction that Theorem 1.1 starts from (p. 5). It says nothing about polynomials other than the maximizer it provides, and decides no degree by itself.