I'm trying to list all roots of a polynomial so I found this paper, in Part 9 on page 29 it gives a simple recipe to find all the roots. But there is this remark:
We have assumed throughout the paper that our polynomial $p$ has all its roots in the unit disk. If instead the roots are in the disk $|z| < r$, you may simply scale all the starting points by a factor of $r$.
My problem is finding some $r$ which matches this. I can see that $r$ is basically any number larger than the maximum magnitude of all the roots. It dosen't matter if it's the smallest $r$, but the smaller the better.
I found a way that I have ben unable to find counter exambles to, albeit the $r$ will be much bigger in some cases, which is having $r$ be the product of the absolute value of all coefficients. (Not counting the polynomial $0$ since anything is a root). But I have no idea how I would go about proving that this $r$ is sufficient.