Problem 1046

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claims/: The 1 claim page of Problem 1046, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f\in \mathbb{C}[x]$ be a monic polynomial and

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

If $E$ is connected then is $E$ contained in a disc of radius $2$?

Status. The site labels the problem DISPROVED, but its commentary answers the stated question yes: "The answer is yes, and in fact the centre of this disc can be taken to be $\frac{z_1+\cdots+z_n}{n}$, where the $z_i$ are the roots of $f$, as shown by Pommerenke [Po59]." Pommerenke's Theorem 3 (Michigan Math. J. 6 (1959), p. 222) states that if $E$ is connected then the lemniscate $|f(z)|=1$ lies in the circle $|z-\zeta|<2$, $\zeta$ the centroid of the zeros, and $E$, bounded by that lemniscate, lies in the same open disc; the paper introduces the theorem as establishing "the conjecture in Problem 14" of [EHP58], the question stated above. So the Statement is proved ([[problems/analysis/E1046/claims/1959_01_01_pommerenke|Pommerenke's claim page]], accepted, full, refereed). The only refutation the commentary reports is of a different conjecture from the same passage of [EHP58], Problem 15, that the width of a connected $\{|f|\le1\}$ is at most $2$: the Remarks of the same paper (pp. 224--225) give $\sup b\ge\sqrt3\,2^{1/3}>2.18$. The site's label fits that conjecture, not the Statement, and the commentary names nothing else as false. The page departs from the site's label here: DISPROVED, the site's "solved in the negative", contradicts Pommerenke's theorem in print and the site's own commentary, and the formal-conjectures statement for the problem is tagged solved with its proof attribute pointing to a Lean proof of the affirmative.

Source. erdosproblems.com/1046, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1046, https://www.erdosproblems.com/1046.

References.

• [EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
• [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. The paper is open access at the publisher. 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 $|f(z)|=1$; the paper introduces it as establishing "the conjecture in Problem 14" of [EHP58], the question stated above, so the stated question is answered yes with the center at the centroid of the zeros, as the site's commentary says. Theorem 4, p. 223, gives $2\le d<4$, $b^2\le32/3$ and $b^2+d^2\le64/3$ for the diameter $d$ and width $b$ of a connected $E$, and the Remarks, pp. 224--225, give the width example the site's commentary reports, $\sup b\ge\sqrt3\,2^{1/3}>2.18$ for the class with $E$ connected, "whereas Erdös, Herzog and Piranian [1, Problem 15] conjectured that $b\le2$ in all cases". The site's DISPROVED label sits beside both statements and names neither; the Status sentence above attaches the standing to the stated question. Library home: pommerenke_1959_some_problems_erdos_herzog_piranian and its theorem_3 and theorem_4 pages.

Formalization. Statement in formal-conjectures, added on 2026-09-22; at its revision of 2026-09-30, erdos_1046 and its centroid variant are tagged solved with their proofs left as sorry and carry formal_proof attributes pointing to the Lean file in Alexeev's repository linked from the claim page above; a diameter variant, that a connected closed sublevel set has diameter at least $2$, is tagged solved without an attribute.

Progress

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Known Results

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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_1959_some_problems_erdos_herzog_piranian / theorem_4
• erdos_1958_metric_properties_polynomials
• erdos_1958_metric_properties_polynomials / problem_14