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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 Milnor number ： ウィキペディア英語版 Milnor number In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If ''f'' is a complex-valued holomorphic function germ then the Milnor number of ''f'', denoted ''μ''(''f''), is either an integer greater than or equal to zero, or it is infinite. It can be considered both a geometric invariant and an algebraic invariant. This is why it plays an important role in algebraic geometry and singularity theory. == Geometric interpretation == Consider a holomorphic complex function germ ''f'': :$f\; :\; (\backslash mathbb^n,0)\; \backslash to\; (\backslash mathbb,0)\; \backslash \; .$ Thus for an ''n''-tuple of complex numbers $z\_1,\backslash ldots,z\_n$ we get a complex number $f(z\_1,\backslash ldots,z\_n).$ We shall write $z\; :=\; (z\_1,\backslash ldots,z\_n).$ We say that ''f'' is singular at a point $z\_0\; \backslash in\; \backslash mathbb^n$ if the first order partial derivatives $\backslash partial\; f\; /\; \backslash partial\; z\_1,\; \backslash ldots,\; \backslash partial\; f\; /\; \backslash partial\; z\_n$ are all zero at $z\; =\; z\_0$. As the name might suggest: we say that a singular point $z\_0\; \backslash in\; \backslash mathbb^n$ is isolated if there exists a sufficiently small neighbourhood $U\; \backslash subset\; \backslash mathbb^n$ of $z\_0$ such that $z\_0$ is the only singular point in ''U''. We say that a point is a degenerate singular point, or that ''f'' has a degenerate singularity, at $z\_0\; \backslash in\; \backslash mathbb^n$ if $z\_0$ is a singular point and the Hessian matrix of all second order partial derivatives has zero determinant at $z\_0$: :$\backslash det\backslash left(\; \backslash frac\; \backslash right)\_^\; =0.$ We assume that ''f'' has a degenerate singularity at 0. We can speak about the multiplicity of this degenerate singularity by thinking about how many points are infinitesimally glued. If we now perturb the image of ''f'' in a certain stable way the isolated degenerate singularity at 0 will split up into other isolated singularities which are non-degenerate! The number of such isolated non-degenerate singularities will be the number of points that have been infinitesimally glued. Precisely, we take another function germ ''g'' which is non-singular at the origin and consider the new function germ ''h := f + εg'' where ''ε'' is very small. When ''ε'' = 0 then ''h = f''. The function ''h'' is called the morsification of ''f''. It is very difficult to compute the singularities of ''h'', and indeed it may be computationally impossible. This number of points that have been infinitesimally glued, this local multiplicity of ''f'', is exactly the Milnor number of ''f''. 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「Milnor number」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.