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Source. The unnumbered Theorem on p. 2 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

Setting (p. 1). For a monic complex polynomial $p$ of degree $n$, the filled lemniscate is $\Lambda_p=\{z\in\mathbb C:\lvert p(z)\rvert<1\}$.

Theorem (p. 2, quoted). "Let $\mathcal P_n(\overline{\mathbb D})$ denote the set of monic polynomials of degree $n$ having all zeros in the closed unit disc. Then for $n\geq3$,

$$\frac{c}{\log n}\leq\inf_{p\in\mathcal P_n(\overline{\mathbb D})}m(\Lambda_p)\leq\frac{C}{\log\log n},$$

where $m(\cdot)$ denotes the two-dimensional Lebesgue measure."

Here $c,C$ are positive finite constants that are pure numbers (the paper's Notation, p. 4). The paper places the result against Pommerenke's 1961 lower bound of order $n^{-4}$ and Wagner's 1988 upper bound of order $(\log\log n)^{-1/2+\delta}$ for every $\delta>0$ (p. 2).

Proof pointer

The paper says (p. 2) that the theorem follows from the finer Theorem 1, which compares the closed-disc constraint with zeros on the unit circle. Theorem 1 is stated for all large enough $n$; the range $n\ge3$ here is the paper's own.

Read depth

Claims checked: the statement was read clause by clause on p. 2 of the print. The proof was not checked.

Bears on

• Problem 116: the lower bound $c/\log n$ is the problem's parenthetical stronger form, an area of at least $(\log n)^{-O(1)}$, with exponent $1$; the paper describes it (p. 2) as an affirmative answer to Erdős's question whether a lower bound of order $(\log n)^{-1}$ holds. The upper bound $C/\log\log n$ shows the minimal area tends to $0$.