Erdős Problem #1047
DISPROVED LEANdatabase updated 2026-01-21
The statement as posed is false. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
The sublevel set \(\{ z: \lvert f(z)\rvert\leq c\}\) of a polynomial \(f\in\mathbb{C}[x]\).
The strict sublevel set \(\{ z: \lvert f(z)\rvert< c\}\) of a polynomial \(f\in\mathbb{C}[x]\).
The connected components of a set `s ⊆ ℂ`, viewed as subsets of `ℂ`.
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? A question of Grunsky, which was reported by Erdős, Herzog, and Piranian [EHP58]. The answer is no, as shown by Pommerenke [Po61], who showed that, if \(k\) is sufficiently large, and \(f(z)=z^k(z-a)\) where \(a\) is sufficiently close to \((1+\frac{1}{k})k^{\frac{1}{k+1}}\), then \(\{ z: \lvert f(z)\rvert\leq 1\}\) has two components, and the component which contains \(0\) is not convex. This was formalized in Lean by Alexeev using Aristotle.
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved · 5 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/1047.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/1047
Attempts
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