= Total ramification of a holomorphic map
{title2=$R_f=\sum_{p\in X}(e_p-1)$}
Count <ramification points> upstairs in $X$, weighted by $e_p-1$, rather than counting distinct <branch values of a holomorphic map> downstairs in $Y$. Grouping by <fibres> gives
$$
R_f=\sum_{y\in Y}\left(d-\#f^{-1}(y)\right).
$$
The cardinality counts distinct points; the <degree of a holomorphic map> counts them with their <analytic ramification indices>. Thus each summand is the number of missing distinct preimages in that <fibre>.<Riemann Surfaces by Constantin Teleman (Lent Term 2003)>[Lecture 5, Definition 5.16, printed p. 19: total branching index as the sum of fibre deficits]{ref}
The phrase <branch point> varies across conventions: to keep the two surfaces distinct, use <ramification point> for $p\in X$ and <branch value of a holomorphic map> for $f(p)\in Y$. The existing name <branch point of a holomorphic map> here denotes the latter. Several <ramification points> can share one <branch value of a holomorphic map>.
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