  An important quantity associated with a complex polynomial $p(z)$ is $\Vert p
\Vert_\infty$, the maximum of its modulus over the unit disc $D$. We prove,
$z_* \in D$ is a local maximum of $|p(z)|$ if and only if $a_*$ satisfies,
$z_*=p(z_*)|p'(z_*)|/p'(z_*)|p(z_*)|$, i.e. it is proportional to its
corresponding Newton direction. This explicit formula gives rise to novel
iterative algorithms for computing $\Vert p \Vert_\infty$. We describe two such
algorithms, including a Newton-like method and present some visualization of
their performance.
