Problem Statement
Let $f\in \mathbb{C}[z]$ be a monic polynomial of degree $n$, all of whose roots satisfy $\lvert z\rvert\leq 1$. Let\[E= \{ z : \lvert f(z)\rvert \leq 1\}.\]What is the shortest length of a path in $E$ joining $z=0$ to $\lvert z\rvert =1$?
Categories:
Analysis
Progress
This is Problem 4.22 in [Ha74], where it is attributed to Erdős. In [Ha74] it is reported that Clunie and Netanyahu (personal communication) showed that a path always exists which joins $z=0$ to $\lvert z\rvert=1$ in $A$.Erdős wrote 'presumably this tends to infinity with $n$, but not too fast'.
The trivial lower bound for the length of this path is $1$, which is achieved for $f(z)=z^n$. The interesting side of this question is what the worst case behaviour is (as a function of $n$).
See also [1041].
Source: erdosproblems.com/1120 | Last verified: January 19, 2026