---

Source. Theorem 7, 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. 4). $\mathcal P_n(K)$ and $\Lambda_p(t)={\lvert p\rvert\le t}$ are as in Theorem 1, and $\Lambda_n(t)$ abbreviates $\Lambda_{p_n}(t)$. For a polynomial $p$ with multiset of zeros $Z$, the empirical measure of zeros is $\mu_p=\frac1{\deg(p)}\sum_{a\in Z}\delta_a$.

Theorem 7 (p. 6, quoted). "Let $K\subseteq\mathbb C$ be a compact set with capacity 1. Let $t>0$ be fixed. Suppose $p_n\in\mathcal P_n(K)$ is a sequence such that $m(\Lambda_{p_n}(t)\cap K)\to0$ as $n\to\infty$. Let $\mu_n$ be the empirical measure of the zeros of $p_n$. Then

$$\mu_n\xrightarrow{w}\nu_K,$$

where $\nu_K$ is the equilibrium measure of $K$."

For $K=\overline{\mathbb D}$ the equilibrium measure is the normalized arc length on the unit circle, so area minimizers at each level $t>0$, whose areas tend to $0$ by the upper bounds of Theorem 1, Theorem 2 and Theorem 3, have zeros that approximately equidistribute on the circle (p. 6).

Proof pointer

Section 7.1 (pp. 25--28). The paper gives two proofs: a compactness argument by potential theory, using the energy-minimizing property of $\nu_K$ and the principle of descent, and a more quantitative one for zeros on the circle. Neither uses optimality, only that the areas vanish.

Read depth

Claims checked: the statement was read clause by clause on p. 6 of the print; the proof was not checked.

Bears on

• Problem 116: background only. Theorem 7 describes the zeros of near-minimizers and is an input to the comparison in Theorem 1; it gives no area bound on its own.