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Subhajit Ghosh and Koushik Ramachandran, Number of components of polynomial lemniscates: a problem of Erdös, Herzog, and Piranian, J. Math. Anal. Appl. 540 (2024), no. 1, Paper No. 128571 (arXiv:2312.13673v1 of 21 December 2023 is the preprint), answer the problem. For a compact $K\subset\mathbb C$ of positive logarithmic capacity $c(K)$ (the transfinite diameter), let $C_n(K)$ be the largest number of connected components of $\{z:|p(z)|<1\}$ over monic $p$ of degree $n$ with all zeros in $K$. Their Question 1.2 restates the problem of Erdős, Herzog and Piranian: is $\limsup_n C_n(K)/n<1$ when $c(K)<1$, and, when $c(K)=1$ and $K$ lies in no closed disc of radius $1$, can $C_n(K)$ equal $n$ along a subsequence of degrees? Theorem 2.1(a) gives the first answer: $0<c(K)<1$ implies $\limsup_n C_n(K)/n<1$, so $C_n(K)\le(1-c)n$ for all large $n$ with a $c>0$ depending on $K$. The paper's notation restricts $K$ to positive capacity, so a closed set of capacity $0$ lies outside the theorem's statement; it is covered all the same, since $C_n$ is monotone in the set and adding a small closed disc $D$ to such a set gives a compact set of capacity $c(D)\in(0,1)$. The paper's remark after the theorem shows that $M(K)=\limsup_n C_n(K)/n$ is not determined by the capacity. The closed disc of radius $1/2$ and the segment $[-1,1]$ both have capacity $1/2$. Every lemniscate over the disc has one component, so $M=0$ there, while the segment has $M>0$. Both sets have $M<1$, so the remark does not decide the parenthetical variant, whether $c$ can be chosen depending only on the capacity. The paper does not treat that variant. Proposition 2.3 gives the second answer: a closed lemniscate $K=Q^{-1}(\overline{\mathbb D})$, which has capacity $1$, has $C_n(K)=n$ for all $n$ in an infinite set. The proposition holds for every closed lemniscate, including the closed unit disc ($Q(z)=z$), which the question's hypothesis excludes; a lemniscate lying in no closed disc of radius $1$, such as $\{z:|z^2-1|\le1\}$, which contains $\pm\sqrt2$, meets the hypothesis and answers the question. Theorem 2.4 gives $\limsup_n C_n(K)/n=1$ for the closure of a bounded Jordan domain with $C^2$ boundary of capacity $1$. Above capacity $1$, Theorem 2.2 gives $C_n(K)=n$ for all large $n$ under a regularity or connectedness condition. The statements above were read from the arXiv preprint (v1), held on its library card; the proofs were not checked here.

Reviewed. The site's curator, Thomas Bloom, marks Problem 1042 proved and credits this paper in the site's commentary (last edited 12 April 2026).

Refereed. Journal of Mathematical Analysis and Applications 540 (2024), no. 1, Paper No. 128571, as the Crossref record of the DOI gives it.