Ramification Index

For a point y in Y, with f(y)=x, the ramification index of f at y is a positive integer e_y such that there is some open neighborhood U of y so that x has only one preimage in U, i.e., f^(-1)(x) intersection U={y}, and for all other points z in f(U), #f^(-1)(z)=e_y. In other words, the map from U to f(U) is e_y to 1 except at y. At all but finitely many points of Y, we have e_y=1. Note that for any point x in X we have sum_(y in f^(-1)(x))=deg(f). Sometimes the ramification index of f at y is called the valency of y.

This entry contributed by Helena Verrill

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Jones, G. A. and Singerman, D. Complex Functions Cambridge, England: Cambridge University Press, p. 196, 1987.

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Ramification Index

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Verrill, Helena. "Ramification Index." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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