Problem #1045: Let $z_1,\ldots,z_n\in \mathbb{C}$ with $\lvert...

---

Progress

A problem of Erdős, Herzog, and Piranian [EHP58], who proved that, for any monic polynomial $f$, if $\{ z: \lvert f(z)\rvert <1\}$ is connected and $f$ has roots $z_1,\ldots,z_n$ then $\prod_{i\neq j}\lvert z_i-z_j\rvert <n^n$.

The value of $\Delta$ when the $z_i$ are the vertices of a regular polygon is $n^n$ when $n$ is even and\[\cos(\pi/2n)^{-n(n-1)}n^n \sim e^{\pi^2/8}n^n\]when $n$ is odd.

Pommerenke [Po61] proved that $\Delta \leq 2^{O(n)}n^n$ for all $z_i$ with $\lvert z_i-z_j\rvert \leq 2$.

Hu and Tang (see the comments) found examples when $n=4$ and $n=6$ that show that vertices of a regular polygon do not maximise $\Delta$. Cambie (also in the comments) showed that, in general, the vertices of a regular polygon are not a maximiser for all even $n\geq 4$.

There is a lot of discussion of this problem in the comments. It is now known that, for even $n$,\[\liminf \frac{\max \Delta}{n^n}\geq C\]for some $C>0$. This was proved with $C\approx 1.0378$ by Sothanaphan [So25]. An alternative construction by Cambie, Dong, and Tang (see the comments by Stijn Cambie) achieves $C\approx 1.304457$ for $6\mid n$, and $C\approx 1.26853$ for all even $n$.


It remains possible that the regular polygon is a maximiser for odd $n$.