Problem 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 $\leq 2$?
Categories:
Analysis Polynomials
Progress
Cartan proved this is true with $2$ replaced by $2e$, which was improved to $2.59$ by Pommerenke [Po61]. Pommerenke [Po59] proved that $2$ is achievable if the set is connected (see [1046]).The generalisation of this to higher dimensions was asked by Erdős as Problem 4.23 in [Ha74].
Source: erdosproblems.com/509 | Last verified: January 15, 2026