Fixed points of complex rational functions dilemma.

Question

There's a theorem about fixed points of complex rational functions which states that those with degree $d$ have $d+1$ fixed points. the case where denominator has greater degree makes sense and it's easy to prove. But how can we show this when numerator has greater degree?
For instance:
$f(z)=\frac{z^3}{z-1}$ then for finding the fixed points: $\frac{z^3}{z-1}=z\implies z^2-z=z^3$ But this will give us only $3$ roots so we only have $3$ fixed points instead of $4$. How is this possible? Where am I doing wrong? How can we prove our theorem?

Answer 1

$\infty$ is a fixed point when the degree of the numerator is greater than the degree of the denominator.

When considering rational functions, one naturally regards them as mappings from the Riemann sphere to itself.