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Let be a monic polynomial with distinct roots, and let be a constant small enough such that has distinct connected components.
Must all these components be convex?
Source: erdosproblems.com/1047
An accepted solution exists. The statement is false.
DISPROVED (LEAN) on the site. Pommerenke's Theorem 14 of 1961, whose closed sublevel set at level has two components, one of them not convex, is the accepted disproof in the problem's exact terms (Pommerenke's claim page (1961)); 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 (1966)); the Lean qualifier of the site's label refers to Alexeev's Lean proof with , 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 (2026)).