Rouche's Theorem and Roots

Question

Use Rouché’s Theorem to find the number of roots of the polynomial $z^5 +z^4 +3$ in the annulus $1 < |z| < 2$.

Answer 1

I'll explain the general gist of Rouche's theorem in a way that can be applied to your problem.

Suppose we have a holomorphic function $h(z)$ defined on a region bounded by a contour $C$, and we want to count how many zeroes $h(z)$ has inside this region. Sometimes, if we're lucky, we can achieve this by splitting up $h(z)$ as a sum, $$ h(z) = f(z) + g(z),$$ where:

• $f(z)$ is a nice, simple holomorphic function, whose zeroes are very easy to count.

• $g(z)$ is small compared to $f(z)$ on the contour $C$. This is to guarantee that $h(z) = f(z) + g(z)$ has the same number of zeroes as $f(z)$ in the region bounded by $C$. According to Rouche's theorem, the precise definition of "smallness" is that $|g(z)| < |f(z)|$ for all $z$ on the contour $C$.

[Notice that, although we're counting zeroes inside the region bounded by $C$, the condition $|g(z)| < |f(z)|$ only needs to be checked on $C$ itself. This is because the number of zeroes of a holomorphic function in the region bounded by $C$ is the same as the number of times its phase "rotates" as we go once around the contour $C$, and if $|g(z)| < |f(z)|$ for all $z \in C$, then the phase of $f(z) + g(z)$ rotates the same number of times as the phase of $f(z)$ as we go around $C$.]

For instance, if $C$ is the unit circle and $h(z) = 4z + e^z$, then we may as well take $f(z) = 4z$ and $g(z) = e^z$. The function $f(z) = 4z$ is nice and simple, and it is easy for us to see that it has one zero inside the unit circle. Also, the function $g(z) = e^z$ is sufficiently small in comparison to $f(z) = 4z$ on the unit circle: indeed, since $|e^z| \leq e < 4 = |4z|$ for all $z$ on the unit circle, the condition $|g(z)| < |f(z)|$ is satisfied for all $z $ on the unit circle. Hence, by Rouche's theorem, we can conclude that $h(z)$ has the same number of zeroes as $f(z)$ inside the unit disk, which is one.

You can apply the same method to your homework problem. It is a good idea to count the number of zeroes in $|z| < 2$, then count the number of zeroes in $|z| < 1$, then take the difference. On each of the contours $|z| = 2$ and $|z| = 1$, you need to find a way of splitting $z^5 + z^4 + 3$ into a "big part" and a "small part", and the "big part" should be something simple enough for you to count its zeroes easily. The way you split up your function will be different for each of the two contours.