Question

Rouche theorem complex analysis number of roots inside unit scirle

Answer 1

Using the Argument Principle and applying Rouche's Theorem, I know that there are 6 zeroes of the polynomial
z10−6z6+3z4−1
z10−6z6+3z4−1
inside the unit circle |z|=1|z|=1, no zeroes inside |z|=1/2|z|=1/2, but I'm not sure how to compute the number of zeroes inside the circle |z|=3/2|z|=3/2.

The best I can do, so far, is to compute the number of zeroes inside the larger circle |z|=2|z|=2. I get 1010 zeroes.

So, I know this polynomial must have 10−6=410−6=4 zeroes ...between |z|=1|z|=1 and |z|=2.