Problem 1042

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claims/: The 1 claim page of Problem 1042, one per claimant's result; the problem's standing derives from them.

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Statement. Let $F\subset\mathbb{C}$ be a closed set of transfinite diameter $1$ which is not contained in any closed disc of radius $1$.

If $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ with all $z_i\in F$ then can

$$\{ z: \lvert f(z)\rvert < 1\}$$

have $n$ connected components?

If the transfinite diameter of $F$ is $<1$ then must this set only have at most $(1-c)n$ connected components, where $c>0$ depends only on $F$ (or just the transfinite diameter of $F$)?

Formulation. The site's second question does not say for which $n$ the bound must hold. Its source, Problem 6 of Erdős, Herzog and Piranian [EHP58, p. 139], asks for a positive $c$, depending on $F$ or perhaps only on its transfinite diameter, such that the set has at most $(1-c)n$ components when $n$ is large. Read for every $n$, the wording fails trivially at $n=1$, where the set is an open disc of radius $1$. This page reads it as the source does, as $\limsup_n C_n(F)/n<1$, which is Ghosh and Ramachandran's Question 1.2. The first question is read as the source, the paper and the site's commentary read it: whether some closed set of transfinite diameter $1$ lying in no closed disc of radius $1$ gives $n$ components for infinitely many $n$. Read for every such set, the question is not answered. The paper proves $n$ components along a subsequence only for closed lemniscates, and it conjectures only the weaker $\limsup_n C_n/n=1$ for every compact set of capacity $1$.

Status. Proved. The site credits Ghosh and Ramachandran [GhRa24]; the standing derives from their claim page.

Source. erdosproblems.com/1042, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1042, https://www.erdosproblems.com/1042.

References.

• [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
• [GhRa24] Ghosh, Subhajit and Ramachandran, Koushik, Number of components of polynomial lemniscates: a problem of Erdös, Herzog, and Piranian. J. Math. Anal. Appl. (2024), Paper No. 128571, 21.

Formalization. None recorded.

Current assessment

The site's formulation asks two things of a closed set $F$ of transfinite diameter (logarithmic capacity) $1$ that lies in no closed disc of radius $1$: whether $\{|f|<1\}$ can have $n$ components for a monic $f$ of degree $n$ with zeros in $F$, and whether capacity below $1$ forces at most $(1-c)n$ components with $c>0$ depending on $F$, or on its capacity alone. Ghosh and Ramachandran answer both. For $0<c(F)<1$ the maximal component count $C_n(F)$ satisfies $\limsup_n C_n(F)/n<1$, so $C_n(F)\le(1-c)n$ for all large $n$ with $c$ depending on $F$; their theorem assumes positive capacity, and a closed $F$ of capacity $0$ is covered all the same, since $C_n$ is monotone in the set and adding a small closed disc $D$ gives a compact $F\cup D$ of capacity $c(D)\in(0,1)$ to which the theorem applies. The paper's remark after Theorem 2.1 shows that $\limsup_n C_n(F)/n$ is not a function of the capacity. The closed disc of radius $1/2$ gives one component for every $n$, and the segment $[-1,1]$, of the same capacity, gives a positive proportion of $n$. Both values are below $1$, so the remark does not decide whether $c$ can depend on the capacity alone, and the paper does not treat that variant. At capacity exactly $1$, a closed lemniscate $Q^{-1}(\overline{\mathbb D})$ attains $C_n=n$ along an infinite set of degrees; this holds for every closed lemniscate, including the closed unit disc ($Q(z)=z$), which the first question excludes, so the question is answered by a lemniscate lying in no closed disc of radius $1$, such as $\{z:|z^2-1|\le1\}$, which contains $\pm\sqrt2$. The closure of a bounded Jordan domain with $C^2$ boundary and capacity $1$ has $\limsup_n C_n/n=1$ (Theorem 2.4). The accepted claim is recorded on [[problems/analysis/E1042/claims/2023_12_21_ghosh_ramachandran|their claim page]] with the journal publication and the curator's credit; the statements are those of the paper, whose library card the claim page links, and the proofs were not checked here.

The first question is answered along a subsequence of degrees, as the paper's own restatement of the problem allows; $C_n=n$ for every large $n$ is proved only above capacity $1$, under a regularity or connectedness condition. Erdős, Herzog and Piranian [EHP58] had shown that the closed unit disc, of capacity $1$, gives $n$ components for every $n$. The site's one thread comment corrects a spelling and claims nothing. Search scope, 2026-10-07: the site page, its thread and the arXiv and Crossref records.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1973_remark_polynomials_transfinite_diameter
• erdos_1973_remark_polynomials_transfinite_diameter / theorem_p23
• ghosh_2024_number_components_polynomial_lemniscates_problem_erdos
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_6
• erdos_1958_metric_properties_polynomials / theorem_7