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Source. Theorem 3, p. 5, of Manjunath Krishnapur, Erik Lundberg and Koushik Ramachandran, On the area of polynomial lemniscates, arXiv:2503.18270v1 (24 March 2025), as identified on the source card.

Statement

The notation $\kappa_n(K,t)$, $\mathbb T$ and $\overline{\mathbb D}$ is that of Theorem 1.

Theorem 3 (p. 5, quoted). "For $t\in(0,1)$, there exist $0<c<C<\infty$ (depending only on $t$) such that for all $n\geq1$,

$$\frac{c}{n^4}\leq\kappa_n(\overline{\mathbb D},t)\leq\kappa_n(\mathbb T,t)\leq\frac{C}{n}.$$

On the other hand, $\kappa_n(\mathbb T,t)\geq\frac{c}{n^2\log n}$."

The upper bound comes from $z^n-1$ (p. 10). Remark 5 (p. 5) adds that for $t=1-\varepsilon_n$ with $\varepsilon_n=\exp(-(\log n)^M)$ the proofs give $\kappa_n(\overline{\mathbb D},t)\le\kappa_n(\mathbb T,t)\lesssim(\log n)^M/n$.

Proof pointer

Upper bound: Section 4 (p. 10), from $p(z)=z^n-1$. Lower bounds: Section 6.3 (pp. 20--23). The bound $c/n^4$ for the closed disc follows an argument of Pommerenke with Bernstein's inequality (p. 20). The bound $c/(n^2\log n)$ for zeros on the circle counts sign changes of $\log\lvert p\rvert$ in small balls, with Wagner's lower estimate for the arc length of $\Lambda(t)\cap\mathbb T$ and covering arguments.

Read depth

Claims checked: the statement was read clause by clause on p. 5 of the print; the proof was not checked.

Bears on

• Problem 116: background only. The problem concerns the level $1$; Theorem 3 concerns fixed levels below $1$, where the minimal area is at most $C/n$.