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Source. Theorem 8, p. 6, 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. 6). The inradius $\rho(\Omega)$ of an open set $\Omega\subseteq\mathbb C$ is the radius of the largest disc contained in $\Omega$. $\mathcal P_n(\overline{\mathbb D})$ is the set of monic degree-$n$ polynomials with all zeros in the closed unit disc, and $\Lambda_p$ is the level-$1$ lemniscate of $p$.

Theorem 8 (p. 6, quoted). "The minimal inradius $\rho_n=\inf\{\rho(\Lambda_p):p\in\mathcal P_n(\overline{\mathbb D})\}$ satisfies

$$\rho_n\geq\frac{c}{n\sqrt{\log n}}."$$

The paper recalls (p. 6) that Erdős, Herzog and Piranian asked whether $\rho_n\ge c/n$ for some $c>0$, and that Pommerenke proved $\rho_n\ge c/n^2$.

Proof pointer

The paper says (p. 6) that the theorem follows from Theorem 1 combined with Lemma 9: the area of the lemniscate is at least $c/\log n$, and the inradius is at least a constant times the square root of the area divided by $n$.

Read depth

Claims checked: the statement was read clause by clause on p. 6 of the print. No separate proof is printed.

Bears on

• Problem 1039: Theorem 8 bounds $\rho(f)$ below by $c/(n\sqrt{\log n})$ for every monic degree-$n$ $f$ with all zeros in the closed unit disc. It does not reach the bound $\rho(f)\gg1/n$ the problem asks about and does not determine the asymptotic behaviour of $\rho_n$; the paper says (p. 3) that it supports the Erdős–Herzog–Piranian conjecture with only the loss of the logarithmic factor.