How to find number of zeros of polynomial $f(z)=z^{13}-2z^7+10z^3-z+4$ in ring $P=\{z:1\leq |z|<2\}$?
Number of zeros in B(0,2) is 13 and there is no zeros in B(0,1). Therefore, number of zeros is 13. Am I right?
Any help is welcome. Thanks in advance.
No, that is incorrect.
On $\lvert z\rvert=1$, we have $g(z)=10z^3$ satisfies $\lvert g(z)\rvert=10$, $\lvert f(z)-g(z)\rvert\leq 8<\lvert g(z)\rvert$. So Rouché's gives $f$ has as many roots as $g$ inside the unit disc with multiplicity, i.e., 3 roots.