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Source. Lemma 9, 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

$\Lambda_p(t)=\{\lvert p\rvert\le t\}$, $\rho$ is the inradius and $m$ the Lebesgue measure, as in Theorem 8.

Lemma 9 (p. 6, quoted). "Let $t>0$. Let $p$ be a degree-$n$ polynomial $p$ (not necessarily monic). Then the inradius $\rho(\Lambda_p(t))$ of its associated lemniscate satisfies

$$\rho(\Lambda_p(t))\geq\frac1{72\pi\sqrt\pi}\frac{\sqrt{m(\Lambda_p(t))}}{n}."$$

Context (p. 3). Cuenya and Levis conjectured in 2005 a constant $C(n)$ depending only on $n$ with $\rho(\Lambda_p)\ge C(n)\sqrt{m(\Lambda_p)}$ for all polynomials $p$ of degree $n$; Solynin and Williams proved it without information on how $C(n)$ depends on $n$ and conjectured that the sharp $C(n)$ is inversely proportional to $n$. Lemma 9 gives $C(n)$ of that form. By Remark 28 (p. 35), the estimate is asymptotically sharp apart from the coefficient $\frac1{72\pi\sqrt\pi}$, as the Erdős lemniscate $\{\lvert z^n-1\rvert<1\}$ shows: its area tends to a constant while its inradius is of order $n^{-1}$.

Proof pointer

Section 8 (pp. 33--35). Lemma 26 gives $A\le18\pi\rho L$ for a bounded simply connected domain with rectifiable boundary of length $L$, area $A$ and inradius $\rho$, used as display (59), and Lemma 27 gives $L(\Lambda)\le4n\sqrt{\pi A(\Lambda)}$ for $\Lambda=\Lambda_p(t)$; the lemma follows from the two.

Read depth

Claims checked: the statement was read clause by clause on p. 6 of the print, and Remark 28 on p. 35; the proof was followed for its structure only.

Bears on

• Problem 1039: with the area bound of Theorem 1 it gives Theorem 8, the bound $\rho_n\ge c/(n\sqrt{\log n})$, short of the problem's $1/n$.