Problem 509

../

claims/: The 3 claim pages of Problem 509, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)\in\mathbb{C}[z]$ be a monic non-constant polynomial. Can the set

$$\{ z\in \mathbb{C} : \lvert f(z)\rvert \leq 1\}$$

be covered by a set of circles the sum of whose radii is $\leq 2$?

Status. Open. The site labels the problem OPEN (page last edited 2025-12-29). Two accepted partial claims, both Pommerenke's, answer yes for the connected case the site's remarks credit to him: his 1959 theorem for every monic $f$ whose open set $\{|f|<1\}$ is connected (Pommerenke 1959), and his 1961 paper for every monic $f$ whose closed set $\{|f|\le1\}$ is connected (Pommerenke 1961). One pending partial claim on the proof-claims tab, recorded here and not adopted: Hong 2026, registered on 2026-07-19 with a write-up in a shared file, which asserts the answer yes for degrees two to four, the quartic case computer-assisted, and credits ChatGPT for most of the text; no review or acceptance of it is recorded (proof-claims thread, 2026-10-07).

Source. erdosproblems.com/509, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #509, https://www.erdosproblems.com/509.

References.

• [Ha74] Hayman, W. K., Research problems in function theory: new problems. (1974), 155-180.
• [Po59] Pommerenke, Ch., On some problems by Erdős, Herzog and Piranian. Michigan Math. J. 6 (1959), no. 3, 221--225, DOI 10.1307/mmj/1028998227. Theorem 3, printed p. 222: "Let $\zeta=(z_1+\cdots+z_n)/n$, where $z_1,\cdots,z_n$ are the zeros of $f(z)$. If $E$ is connected, then $C$ is contained in the circle $|z-\zeta|<2$", $C$ the lemniscate $|f(z)|=1$ and $E=\{|f(z)|<1\}$; this is the result behind the site's remark that the constant $2$ can be reached when the set is connected: one disc of radius $2$ covers $\{|f(z)|\le1\}$ when the open set $E$ is connected (the site's remark names the closed set, a weaker condition that Pommerenke 1961 covers in the proof of Theorem 10(b), p. 107); the accepted partial claim on Pommerenke 1959. The paper does not treat the general case of this page's question. Library home: pommerenke_1959_some_problems_erdos_herzog_piranian and its theorem_3 page.
• [Po60] Pommerenke, Ch., Einige Sätze über die Kapazität ebener Mengen. Math. Ann. 141 (1960), 143--152, DOI 10.1007/BF01360168. Satz 3 (p. 149): a closed set $E$ of positive capacity is covered by finitely many discs whose radii sum to less than $2.59\operatorname{cap}E$. It follows from Satz 2, under which finitely many curves of total length below $10.36\operatorname{cap}E$ enclose $E$. The introduction (p. 143) applies it to the set where a monic polynomial of degree $n$ has modulus at most $r^n$, whose capacity is $r$, as a sharpening of the Boutroux--Cartan lemma. The site's commentary credits the bound $2.59$ to Pommerenke under its key [Po61], which its reference record resolves to the 1961 paper below. That paper cites this one as its [12], for the length bound in its Theorem 7, and none of its numbered statements gives the covering bound.
• [Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; Theorem 10 and the containment of a connected $E$ in the disk of radius 2 about the centroid, printed pp. 106--107, the accepted partial claim on Pommerenke 1961; Theorem 2, p. 98; Theorem 6, p. 102. Library home: pommerenke_1961_metric_properties_complex_polynomials (result page theorem_10).

Formalization. Statement in formal-conjectures; at that commit the file states the question and three solved variants (Cartan's bound, Pommerenke's bound and the connected case) without proofs.

Current assessment

The general question is open. The known general bounds settle no instance of it: Cartan's lemma covers $\{|f|\le1\}$ by discs whose radii sum to $2e$, and Pommerenke [Po60] lowers that constant to $2.59$ (Satz 3, as Eremenko and Hayman (1999) also cite it; the site cites [Po61]; see References), while the question asks for the sum $2$. Pommerenke [Po59] proves that when the open set $\{|f|<1\}$ is connected, the lemniscate $|f|=1$ lies in the open disc of radius $2$ about the centroid of the zeros, so one disc of radius $2$ covers $\{|f|\le1\}$; that answers yes for every monic $f$ with connected $\{|f|<1\}$ and is the accepted partial claim on Pommerenke 1959. Pommerenke [Po61] extends it to the case the site's remark names: when the closed set $\{|f|\le1\}$ is connected, it lies in the closed disc of radius $2$ about the centroid (proof of Theorem 10(b), p. 107), the accepted partial claim on Pommerenke 1961. Both are refereed in the Michigan Mathematical Journal and credited by the site's remarks, which label the problem OPEN. The pending partial claim on Hong 2026 asserts the answer yes for degrees two to four, with a write-up behind a sign-in and no review. Erdős asked the higher-dimensional generalization as Problem 4.23 of [Ha74]. Search scope: the site's problem page and proof-claims tab (2026-10-07), the formal-conjectures statement file, and the two Pommerenke papers filed in the library; no other literature search is recorded.

Known Results

• Cartan's lemma: $\{|f|\le1\}$ is covered by discs whose radii sum to $2e$; Pommerenke [Po60] improves the constant to $2.59$ (Satz 3; the site cites [Po61], which contains no such bound). Neither reaches $2$, so neither settles an instance of the question.
• [Po59], Theorem 3: when $\{|f|<1\}$ is connected, one disc of radius $2$ about the centroid of the zeros covers $\{|f|\le1\}$; accepted partial claim on Pommerenke 1959.
• [Po61], proof of Theorem 10(b): when $\{|f|\le1\}$ is connected, it lies in the closed disc of radius $2$ about the centroid of the zeros; accepted partial claim on Pommerenke 1961.
• Claimed, not accepted: the answer yes for monic $f$ of degree two, three or four, on Hong 2026.

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.

• pommerenke_1959_some_problems_erdos_herzog_piranian
• pommerenke_1959_some_problems_erdos_herzog_piranian / theorem_3
• pommerenke_1961_metric_properties_complex_polynomials
• pommerenke_1961_metric_properties_complex_polynomials / example_p103
• pommerenke_1961_metric_properties_complex_polynomials / theorem_10
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / claim_3_1
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / corollary_4_3
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / lemma_4_1
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / lemma_4_2
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / lemma_5_1
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / lemma_7_1
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / proposition_6_1
• hong_2026_strategy_proposal_covering_lemniscates_erdos_problem / section_8_4