Covering a polynomial lemniscate by discs

Show that the set where a monic polynomial has modulus at most 1 can be covered by discs whose radii sum to at most 2.

shoal State the problem, from Erdős Problems 5ed1837 · 2周前历史

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1	# Covering a polynomial lemniscate by discs
2
3	For a monic polynomial $f$, the region where $\lvert f(z)\rvert\le1$ can be one blob or many scattered pieces around the roots. Erdős asked whether it can always be covered by discs whose radii add up to at most $2$. Cartan's classical lemma gives the constant $2e$, Pommerenke improved it to $2.59$, and the sharp constant $2$ is known only when the region is connected.
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5	## Problem
6
7	Let $f(z)\in\mathbb{C}[z]$ be a monic polynomial of degree at least $1$, and let
8
9	$$
10	K_f=\{z\in\mathbb{C} : \lvert f(z)\rvert\le1\}.
11	$$
12
13	Can $K_f$ always be covered by finitely many closed discs whose radii have sum at most $2$?
14
15	## Background
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17	Sets of the form $\{\lvert f\rvert\le1\}$ are polynomial lemniscates; Erdős, Herzog and Piranian studied their metric properties in a 1958 paper that contains many related questions. Coverings of this kind are provided by Cartan's lemma, a standard tool for bounding polynomials away from their zeros. Erdős included the question in his 1961 list of problems, and later asked, as Problem 4.23 in Hayman's collection of research problems in function theory, for a generalisation to higher dimensions.
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19	## What is known
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21	- Cartan's lemma gives a covering with total radius at most $2e\approx5.44$.
22	- Pommerenke (1961) improved the constant to $2.59$.
23	- Connected case. Pommerenke (1959) proved that the constant $2$ is achievable when $K_f$ is connected; in fact a connected lemniscate of this kind lies in a single disc of radius $2$ centred at the mean of the roots.
24	- A 2026 attempt posted on the problem's forum proposes a covering built from the connected components of the lemniscate, and a proof claim posted in July 2026 covers polynomials of degree at most $4$, partly by computer; neither has been verified.
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26	## What would count as progress
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28	- A proof that the constant $2$ works in general, or any constant below $2.59$.
29	- A monic polynomial whose lemniscate cannot be covered by discs with total radius $2$; for explicit polynomials the lemniscate can be computed and candidate coverings checked numerically with rigorous error bounds.
30	- Results for special families, for example polynomials with all roots real, or lemniscates with two components.
31	- A formal proof against the Lean statement in [formal-conjectures](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/509.lean).
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33	## References
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35	1. P. Erdős, F. Herzog and G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125–148. [doi:10.1007/BF02790232](https://doi.org/10.1007/BF02790232)
36	2. Ch. Pommerenke, On some problems by Erdős, Herzog and Piranian, Michigan Math. J. 6 (1959), 221–225. [doi:10.1307/mmj/1028998227](https://doi.org/10.1307/mmj/1028998227)
37	3. Ch. Pommerenke, On metric properties of complex polynomials, Michigan Math. J. 8 (1961), 97–115. [doi:10.1307/mmj/1028998561](https://doi.org/10.1307/mmj/1028998561)
38	4. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, Springer (2019). [doi:10.1007/978-3-030-25165-9](https://doi.org/10.1007/978-3-030-25165-9)
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40	## Source
41
42	Imported from [Erdős Problems 509](https://www.erdosproblems.com/509), curated by Thomas Bloom. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.