Problem 114 — Visual explainer (undergrad)
==========================================

Problem statement
---------------
If $p(z)\in\mathbb{C}[z]$ is a monic polynomial of degree $n$ then is the length of the curve $\{ z\in \mathbb{C} : \lvert p(z)\rvert=1\}$ maximised when $p(z)=z^n-1$?

Picture
-------
[Given objects + constraints]  --->  [Count / structure]  --->  [Show bound / existence]

Question: what to look at
------------------------
- Restate the problem as: given the constraint, what asymptotic bound or
  structure must follow?
- Use the comments as benchmarks (best known bounds / constructions).

Benchmarks / known results (from comments)
----------------------------------------
- Let the maximal length of such a curve be denoted by $f(n)$.
- {UL} {LI}The length of the curve when $p(z)=z^n-1$ is $2n+O(1)$, and hence
  the conjecture implies in particular that $f(n)=2n+O(1)$.{/LI} {LI}Dolzhenko
  proved $f(n) \leq 4\pi n$, but few were aware of this work.{/LI}
  {LI}Pommerenke proved $f(n)\ll n^2$.{/LI} {LI}Borwein proved $f(n)\ll n$
  (Borwein was unaware of Dolzhenko's earlier work).
- The prize of \$250 is reported by Borwein.{/LI} {LI}Eremenko and Hayman
  proved the full conjecture when $n=2$, and $f(n)\leq 9.173n$ for all
  $n$.{/LI} {LI}Danchenko proved $f(n)\leq 2\pi n$.{/LI} {LI}Fryntov and
  Nazarov proved that $z^n-1$ is a local maximiser, and solved this problem
  asymptotically, proving that\[f(n)\leq 2n+O(n^{7/8}).\]{/LI} {LI} Tao has
  proved that $p(z)=z^n-1$ is the unique (up to rotation and translation)
  maximiser for all sufficiently large $n$.
- {/UL} Erd\H{o}s, Herzog, and Piranian also ask whether the length is at
  least $2\pi$ if $\{ z: \lvert f(z)\rvert<1\}$ is connected (which $z^n$
  shows is the best possible).