Critical multiplicity of a rational map ($m_p(R)=e_p(R)-1$)

At a point of local degree $e$, the critical multiplicity is $e-1$. At an ordinary finite point this is the order of vanishing of the derivative. The invariant local-coordinate definition also works at poles and infinity. For $R(z)=z^d$, zero and infinity both have multiplicity $d-1$, so a critical-point count must count multiplicities: there are two distinct critical points but total multiplicity $2d-2$.

A degree-$d$ rational map has total critical multiplicity $2d-2$. Here is an algebraic proof. Critical points are isolated zeros of local derivatives and are finite in number by compactness. Change domain and range Möbius coordinates so that the poles are $d$ simple finite poles and infinity is a regular point with a finite image. Write $R=P/Q$, with $\mathrm{deg}Q=d$, and $R(z)=a+b/z+O(z^{-2})$, $b\ne 0$. Then $W=P^{\prime }Q-P{Q}^{\prime }$ has degree exactly $2d-2$, since $R^{\prime }=W/Q^2\sim -b/z^2$. At a pole, $W=-P{Q}^{\prime }\ne 0$. Thus the roots of $W$ are precisely the critical points and their zero orders are the critical multiplicities. The fundamental theorem of algebra gives the count; the coordinate changes preserve it.