Problem Statement
Let $f(z)=\prod_{i=1}^n(z-z_i)\in \mathbb{C}[z]$ with $\lvert z_i\rvert < 1$ for all $i$.
Must there always exist a path of length less than $2$ in\[\{z: \lvert f(z)\rvert < 1\}\]which connects two of the roots of $f$?
Must there always exist a path of length less than $2$ in\[\{z: \lvert f(z)\rvert < 1\}\]which connects two of the roots of $f$?
Categories:
Analysis Polynomials
Progress
A problem of Erdős, Herzog, and Piranian [EHP58], who proved that this set always has a connected component containing at least two of the roots.Source: erdosproblems.com/1041 | Last verified: January 19, 2026