Problem 1120

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 $\lvert z\rvert =1$ ?

Source: erdosproblems.com/1120

Analysis

Wiki page

Status

Open

No claim settles this problem.

The site labels the problem OPEN (page last edited 30 December 2025). Two pending partial claims by Pendyala bear on it: an arXiv preprint asserting $c\sqrt{\log n}\leq S(n)\leq \pi n$ for large $n$ (claim page (Pendyala, 2026)), and an SSRN preprint asserting $S(n)=1$ for $n\leq 3$ and $S(6)>1$ (claim page (Pendyala, *Radial, 2026)).

Problem 1120

Status

Proof claims

Discussion