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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 Local parameter ： ウィキペディア英語版 Local parameter In the geometry of complex algebraic curves, a local parameter for a curve ''C'' at a smooth point ''P'' is just a meromorphic function on ''C'' that has a simple zero at ''P''. This concept can be generalized to curves defined over fields other than $\backslash mathbb$ (or even schemes), because the local ring at a smooth point ''P'' of an algebraic curve ''C'' (defined over an algebraically closed field) is always a discrete valuation ring.〔J. H. Silverman (1986). ''The arithmetic of elliptic curves''. Springer. p. 21〕 This valuation will endow us with a way to count the order (at the point ''P'') of rational functions (which are natural generalizations for meromorphic functions in the non-complex realm) having a zero or a pole at ''P''. Local parameters, as its name indicates, are used mainly to properly ''count multiplicities'' in a local way. ==Introduction== When ''C'' is a complex algebraic curve, we know how to count multiplicities of zeroes and poles of meromorphic functions defined on it.〔R. Miranda (1995). ''Algebraic curves and Riemann surfaces''. American Mathematical Society. p. 26〕 However, when discussing curves defined over fields other than $\backslash mathbb$, we do not have access to the power of the complex analysis, and a replacement must be found in order to define multiplicities of zeroes and poles of rational functions defined on such curves. In this last case, we say that the germ of the regular function $f$ vanishes at $P\backslash in\; C$ if $f\backslash in\; m\_P\backslash subset\backslash mathcal\_$. This is in complete analogy with the complex case, in which the maximal ideal of the local ring at a point ''P'' is actually conformed by the germs of holomorphic functions vanishing at ''P''. Now, the valuation function on $\backslash mathcal\_$ is given by :$\backslash operatorname\_P(f)=\backslash max\backslash ;$ this valuation can naturally be extended to ''K''(''C'') (which is the field of rational functions of C) because it is the field of fractions of $\backslash mathcal\_$. Hence the idea of ''having a simple zero at a point P'' is now complete: it will be a rational function $f\backslash in\; K(C)$ such that its germ falls into $m\_P^d$, with ''d'' at most 1. This has an algebraic resemblance with the concept of a uniformizing parameter (or just uniformizer) found in the context of discrete valuation rings in commutative algebra; a uniformizing parameter for the DVR (''R, m'') is just a generator of the maximal ideal ''m''. The link comes from the fact that a local parameter at ''P'' will be a uniformizing parameter for the DVR ($\backslash mathcal\_$, $m\_P$), whence the name. 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「Local parameter」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.