Huang 2025 many lemniscates large diameter

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theorem_1_1: Huang's main theorem: for every c strictly between 0 and 4 and every positive integer N, some monic polynomial has a closed sublevel set at level one with at least N connected components of diameter at least c.

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Linhang Huang, Many lemniscates with large diameter. arXiv:2509.11597 (2025). The copy read for this card is version 2 (16 September 2025). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2509.11597), every other right reserved.

Theorem 1.1 (p. 1; proof pp. 2--4) shows that for every $c\in(0,4)$ and every $N\in\mathbb N$ there is a monic polynomial $p$, of some degree $n$, such that the closed sublevel set $\{z : |p(z)|\le1\}$ has at least $N$ connected components each of diameter at least $c$. The paper presents this (p. 1) as an answer to the question of Erdős whether the number of components of diameter greater than $1+c$ is bounded by a universal constant $A(c)$ independent of the degree. It calls the restriction $c<4$ best possible by Pólya's theorem that, for a monic polynomial, the orthogonal projection of the sublevel set onto any line can be covered by intervals of total length at most $4$; it adds (p. 2) that a segment of length $\ell$ has logarithmic capacity $\ell/4$, which suggests $4$ as the limit.

The method works with logarithmic capacity (Section 2, pp. 2--4): one explicit domain $\Omega$ of logarithmic capacity $1$ containing $[0,c]$ is built from a shifted Joukowski map, $N$ pairwise disjoint Jordan domains of diameter greater than $c$ are placed inside it, and the Hilbert Lemniscate Theorem with capacity estimates gives a polynomial whose sublevel set contains their union and lies inside $\Omega$, with leading coefficient of modulus at least $1$; a rescaling then makes it monic. A note added on p. 2 says the problem had already been solved by Pommerenke (Michigan Math. J. 8 (1961)), so the paper is an independent rediscovery, kept on arXiv and not submitted to a journal.

Read status: claims checked. Theorem 1.1 was read clause by clause on p. 1 of version 2, and the proof of Section 2 was read through once; nothing is independently reviewed.

Source: https://arxiv.org/abs/2509.11597.

Bears on. #511: the paper states (p. 1) that Theorem 1.1 answers the question of Erdős it identifies as problem #511. The theorem is stated for the closed set $\{|p|\le1\}$ and diameters at least $c$, while the problem's statement uses $|f|<1$ and diameters greater than $c$, for $c>1$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.