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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Riemann–Hurwitz formula ： ウィキペディア英語版 Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves. == Statement == For an orientable surface ''S'' the Euler characteristic χ(''S'') is :$2-2g\; \backslash ,$ where ''g'' is the genus (the ''number of handles''), since the Betti numbers are 1, 2''g'', 1, 0, 0, ... . In the case of an (''unramified'') covering map of surfaces :$\backslash pi:S\text{'}\; \backslash to\; S\; \backslash ,$ that is surjective and of degree ''N'', we should have the formula :$\backslash chi(S\text{'})\; =\; N\backslash cdot\backslash chi(S).\; \backslash ,$ That is because each simplex of ''S'' should be covered by exactly ''N'' in ''S''′ — at least if we use a fine enough triangulation of ''S'', as we are entitled to do since the Euler characteristic is a topological invariant. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (''sheets coming together''). Now assume that ''S'' and ''S′'' are Riemann surfaces, and that the map π is complex analytic. The map π is said to be ''ramified'' at a point ''P'' in ''S''′ if there exist analytic coordinates near ''P'' and π(''P'') such that π takes the form π(''z'') = ''z''''n'', and ''n'' > 1. An equivalent way of thinking about this is that there exists a small neighborhood ''U'' of ''P'' such that π(''P'') has exactly one preimage in ''U'', but the image of any other point in ''U'' has exactly ''n'' preimages in ''U''. The number ''n'' is called the ''ramification index at P'' and also denoted by ''e''''P''. In calculating the Euler characteristic of ''S''′ we notice the loss of ''eP'' − 1 copies of ''P'' above π(''P'') (that is, in the inverse image of π(''P'')). Now let us choose triangulations of ''S'' and ''S′'' with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then ''S′'' will have the same number of ''d''-dimensional faces for ''d'' different from zero, but fewer than expected vertices. Therefore we find a "corrected" formula :$\backslash chi(S\text{'})\; =\; N\backslash cdot\backslash chi(S)\; -\; \backslash sum\_\; (e\_P\; -1)$ or as it is also commonly written :$2g(S\text{'})-2\; =\; N\backslash cdot(2g(S)-2)\; +\; \backslash sum\_\; (e\_P\; -1)$ (all but finitely many ''P'' have ''eP'' = 1, so this is quite safe). This formula is known as the ''Riemann–Hurwitz formula'' and also as Hurwitz's theorem. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Riemann–Hurwitz formula」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.