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Talk:Riemann–Hurwitz formula

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Examples needed

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This page should include examples

A possible mistake in the formula

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It seems to me that the formula and its proof are wrong as stated, since the fiber of a branch point may contain unramified points, and thus the restriction of to the unramified points as stated in the proof is undefined. Any univariate polynomial whose derivative has more than one distinct root provides a counterexample when considered as a map of Riemann spheres.

To correct the error, it seems sufficient to replace by the size of the union of the fibers of the branch points, which contains all the ramification points together with perhaps some unramified points, denote this quantity by . In this formulation, the proof works as stated in the article, and we can also deduce it from the usual formulation of Riemann-Hurwitz, as we have since for any we have Horrific Necktie (talk) 09:36, 12 August 2022 (UTC)[reply]