Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1047

claims/: The 3 claim pages of Problem 1047, one per claimant's result; the problem's standing derives from them.

Statement. Let $f\in \mathbb{C}[x]$ be a monic polynomial with $m$ distinct roots, and let $c>0$ be a constant small enough such that ${ z: \lvert f(z)\rvert\leq c}$ has $m$ distinct connected components.

Must all these components be convex?

Status. DISPROVED (LEAN) on the site. Pommerenke's Theorem 14 of 1961, $z^p(z-a)$ whose closed sublevel set at level $1$ has two components, one of them not convex, is the accepted disproof in the problem's exact terms (Pommerenke's claim page); Goodman's 1966 quartics, one with four simple roots, disprove Grunsky's question for the open sublevel set at a critical level, and the paper does not treat the problem's closed set (Goodman's claim page); the Lean qualifier of the site's label refers to Alexeev's Lean proof with $z^6-z$ , found by Aristotle from the informal statement, which this corpus built at a pinned commit and accepted as a formalized disproof, resting on the classical fact that every component of the set contains a root (Alexeev's claim page).

Source. erdosproblems.com/1047, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1047, https://www.erdosproblems.com/1047.

References.

[EHP58] Erdős, P. and Herzog, F. and Piranian, G., Metric properties of polynomials. J. Analyse Math. (1958), 125-148.
[Go66] Goodman, A. W., On the convexity of the level curves of a polynomial. Proc. Amer. Math. Soc. (1966), 358-361.
[Po61] Pommerenke, Ch., On metric properties of complex polynomials. Michigan Math. J. 8 (1961), no. 2, 97--115, doi:10.1307/mmj/1028998561; Grunsky's question as recalled, pp. 111--112; Theorem 14, p. 112. Library home: pommerenke_1961_metric_properties_complex_polynomials (result page theorem_14).

Formalization. Statement in formal-conjectures, at its revision of 2026-09-18: it states the problem as erdos_1047, with its proof left as sorry and a formal_proof attribute pointing to Alexeev's development, which was built and checked here and is recorded on the claim page above; its variants restate Pommerenke's, Goodman's and the referee's examples, and the variant max_non_convex_components, on the greatest number of nonconvex components by degree (Goodman's follow-up question rather than this one), carries its own formal_proof attribute pointing to a later Lean development pinned to a commit.

Progress

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Known Results

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Linked library material

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Linked from (12)

problems/analysis/E1047/claims Analysis Analysis analysis/goodman_1966_convexity_level_curves_polynomial First counterexample, p. 359: (z^2 + 1)(z - 2)^2 at the level 5 sqrt 5 / 4 Referee's example, p. 361: z(z^5 - 1) at the level 5 / 6^{6/5}Theorem (p. 361): a quartic with four simple roots and a nonconvex component of E(c)analysis/pommerenke_1961_metric_properties_complex_polynomials Theorem 14: z^p (z - a) with two components, one not convex polynomials/erdos_1958_metric_properties_polynomials Problem 16 (Grunsky): are the m loops of a small-level lemniscate all convex?Theorem 11: E is convex when all zeros lie in a disk of radius sin(π/8)/(1+sin(π/8))

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