Given two polynomials $p,q$ where $$\{z: |p(z)|=1\} = \{z:|q(z)| = 1\}$$ show that $p=\xi^n,q=c\xi^m$ for some polynomial $\xi$ and $|c| = 1$.

Solution. (Ang Yan Sheng) Take $r = \frac{p^{\deg q}}{q^{\deg p}}$, and $\{|r(z)| = 1\}$ is also the same locus.

The locus breaks up $\mathbb C$ into several connected components, but exactly one is unbounded (because $p(z) \to \infty$ when $z\to \infty$).

In this unbounded component $U$ (with the “boundary” removed), there are no roots of $p$, since $p(\infty) = \infty$. (Similarly, no roots of $q$ either.)

Therefore, the meromorphic function has no roots or poles in $U$. By applying the maximum modulus theorem to $r$ and $1/r$, we get that both the maximum and minimum modulus of $r$ on $\overline U$ lies on $\partial U$, where $|r| = 1$. So we conclude that $|r|=1$ for all of $U$, so $r$ is constant.