$z$ is in the complex plane, r is real, and P(z) is an nth order polynomial in z
Is there a way to determine the minimum value of $r$ for which the region $|P(z)|<r$ is a connected region? By connected I mean that any point in the region can be reached from any other point in the region by a path entirely in the region.
An equivalent question would be: Is there a way to determine at what value of $r$ the surface $P(z)=r$ changes from being a single continuous surface to multiple disconnected surfaces?
In the limit of large $r$, the region becomes one large circle. In the limit of small $r$, it becomes a set of disconnected circles, one around each root of $P(z)$.
If the threshold $r$ cannot be determined analytically for a general $P(z)$, are there non-trivial special cases of $P(z)$ for which it can?
is not connected.
is further disconnected.