= Critical multiplicity of a rational map
{title2=$m_p(R)=e_p(R)-1$}
= Critical multiplicity
{synonym}
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 $2d-2$.
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