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Source. Theorem 2, 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: the infimum of the area of $\{\lvert p\rvert\le t\}$ over monic degree-$n$ polynomials with all zeros in $K$.

Theorem 2 (p. 5, quoted). "For $t>1$, there exist $0<c<C<\infty$ (depending only on $t$) such that for all large enough $n$,

$$\frac{c}{\log\log n}\leq\frac13\kappa_{n(\log\log n)^4}(\mathbb T,t)\leq\kappa_n(\overline{\mathbb D},t)\leq\kappa_n(\mathbb T,t)\leq\frac{C}{\log\log n}."$$

So for each fixed level above $1$ the minimal area has the sharp order $(\log\log n)^{-1}$. Remark 5 (p. 5) adds that the proofs also give $\kappa_n(\mathbb T,t)\ge\kappa_n(\overline{\mathbb D},t)\gtrsim1/\log\log n$ for $t=1+\varepsilon_n$ with $\varepsilon_n=\exp(-(\log n)^M)$.

Proof pointer

Upper bound: Section 4 (pp. 10--14). Lower bound for zeros on the circle: Section 6.2 (pp. 19--20), by bounding the doubling exponent of $\log\lvert p_n\rvert$ for a minimizer. The passage to the closed disc uses the zero-pushing lemma of Section 7.

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 2 concerns fixed levels $t>1$ and does not bound the level-$1$ area.