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Source. Theorem 6, 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 $\kappa_n(K,t)$ 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 $K$. Capacity is logarithmic capacity throughout the paper (p. 5).

Theorem 6 (p. 5, quoted). "Let $K$ be the closure of a bounded open set having $C^2$-smooth boundary. Assume that $K$ has capacity 1. Then, $\inf_n\kappa_n(K,1)=0$."

The theorem gives an infimum over $n$, not a limit; the paper says (p. 9) that in adapting the disc argument it loses quantitative control of the area in terms of the degree. The abstract describes the result as showing that the minimal area converges to zero as $n\to\infty$. For the context the paper gives (p. 5): Erdős, Herzog and Piranian showed that $\kappa_n(K,1)$ is bounded below by a positive constant when $K$ has capacity less than $1$ and asked whether $\kappa_n(K,1)\to0$ when the capacity is at least $1$; the authors' earlier paper showed $\kappa_n(K,1)\le e^{-cn}$ when the capacity exceeds $1$ and $K$ is sufficiently smooth.

Proof pointer

Section 5 (pp. 14--18). The set may be taken simply connected by filling in bounded complementary components; the construction parallels the one for the disc in Section 4, with a harmonic function built from an entire function and some explicit steps replaced by functional-analytic arguments. Proposition 14 (p. 17) gives quantitative bounds when $K$ is itself a closed unit lemniscate.

Read depth

Claims checked: the statement was read clause by clause on p. 5 of the print; the proof was followed for its structure only.

Bears on

• Problem 1040: for a set $K$ of this kind with transfinite diameter (logarithmic capacity) $1$, Theorem 6 gives $\mu(K)=0$, the infimum in the problem being taken over all degrees. This is the case of transfinite diameter exactly $1$ of the problem's second question, restricted to closures of bounded open sets with $C^2$-smooth boundary. The paper measures $\{\lvert p\rvert\le1\}$ and the problem $\{\lvert f\rvert<1\}$.