I am still unsure how exactly one applies Rouche's Theorem to find the roots of polynomials. For example, to find how many roots $z^9+z^5-8z^3+2z+1$ has in between the circles $|z|=1$ and $|z|=2$. I know we want to divide this into a "big" part and a "small" part, where we easily know the roots of the big part. On this domain wouldn't $z^9$ dominate? This function has no roots on this domain, so the whole function should't, right? I clearly have to be missing something because apparently there are six roots on this domain.
[Math] Using Rouche’s theorem to find number of roots.
complex-analysisroots
Best Answer
First apply Rouche's theorem to find the number of zeros inside the unit disk. Here you need a function which dominates on the boundary of the unit disk and whose zeros inside you know. Note that on the boundary (when $|z|=1$) $-8z^3$ is quite large compared to the other terms of your polynomial $z^9+z^5-8z^3+2z+1$. You should get that there are 3 roots in the disk of radius 1.
Next apply Rouche's theorem on the disk of radius 2. Here you need a function which dominates when $|z|=2$ and whose zeros inside the disk are known. Note that $z^9$ is quite large when $|z|=2$. You should find there are 9 zeros in the disk of radius 2.
Now for the final answer, subtract the number of zeros in the disk of radius 1 from the number of zeros in the disk of radius 2.