The idea is that the zero or pole at infinity cancels out the finite zeroes and poles. For example, in your $quadratic/cubic$ example, there are two finite zeroes, three finite poles, and a simple zero at infinity. The proof is as follows: If $f(z)=\dfrac{z^2+az+b}{z^3+cz^2+dz+e}$, then $f(1/z)=\dfrac{1+az+bz^2}{1+cz+dz^2+ez^3}z$ which clearly has a simple zero at $z=0$. This proof generalizes easily to any rational function, and thus any meromorphic function on the Riemann sphere.
This is a technique to solve the more general problem of counting the number of zeros of a polynomial inside the unit circle. One could use it for other curves other than the circle. All is needed is to be able to map it to a line by a rational function.
The idea is to use the argument principle instead: The number of zeros of a polynomial lying inside a loop is the number of times the image of that loop by the polynomial winds around the origin. But the unit circle is hard on additions. That is why a pretty proof by Rouche's can be tricky sometimes.
Let's instead map the unit circle to the imaginary line.
You might know a rational function that does the map, but we can derive it step by step.
- Translate the circle one unit to the right. $z= x-1$.
- Then we do inversion. Inversion would be $x = 1/\overline{y}$. But since the coefficients are real the conjugate won't matter. So we do $x=1/y$. We get a rational function of which we only care about the numerator (a polynomial). If zero is a solution, then $-1$ was a solution of the original polynomial and that we should've tested before hand. After this the circle got mapped to the vertical line passing through (1/2,0).
- Finally we translate to the left by 1/2. y = w+1/2.
So, we get some polynomial with real coefficients. Let's evaluate it at $w = ir$ with $r$ real.
Now, separate imaginary part and real part. Both a polynomials of smaller degree. For this particular problem I think we get
$$(880r^4-392r^2+23)+r(96r^4-912r^2+54)i$$
Now, to determine the number of times this winds around the origin we just need to see how it jumps from quadrant to quadrant. The counting of roots (no need of precise determination) can be done with Sturm's theorem in general.
For this particular problem the work is much easier. For $r=0$ we are at the point (23,0). The polynomials $880r^4-392r^2+23$ and $96r^4-912r^2+54$ are just quadratics in disguise. One can compute the roots if so inclined.
But all it matters is their relative position, which I think it is $ABBAABBA$, where the $A$'s represent roots from the second polynomial and the $B$'s represent the roots of the first one. Take into account the factor $r$ in the imaginary part which also changes its sign when $r$ crosses zero.
That order of the roots tells you the sign of the imaginary part and real part on each of the intervals between the roots. This tells you to which quadrant the whole expression is moving. From the succession of quadrants you count the winding number and that is your number.
Best Answer
The two equations don't have the same number of solution in the given sector for every $k$. Define $(I)$ as equation $z^k+1=0$ and $(II)$ as equation $(1+z^k)(1+z)^k+z^k=0$. For $k=8$, numerically $(I)$ has $1$ solutions and $(II)$ has no solutions in sector $\frac{\pi}{2} < \text{Arg}(z) < \frac{2\pi}{3}$. If you denote $N_1,N_2$ as the solutions of $(I),(II)$ respectively in the given sector, then for $k\le 50$, (all of them are even :o )
$$ \begin{array}{c|c|c} \hline k& N_1& N_2\\ \hline 8& 1& 0\\ \hline 10& 0& 1\\ \hline 14& 1& 0\\ \hline 18& 1& 2\\ \hline 20& 2& 1\\ \hline 32& 3& 2\\ \hline 34& 2& 3\\ \hline 38& 3& 2\\ \hline 42& 3& 4\\ \hline \end{array} $$
Now coming to magnitude of the solutions, all solutions to the right of $x=-1/2$ line are indeed close to each other. This is illustrated in the image below for $k=195$.
Addendum: The following matlab code was used to produce this image, it only works till $k=195$:
Observe that some zeroes of equation $II$ are vertically stacked near the vertical line $x=-1/2$. This is also observed for smaller values of $k$,