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In mathematics, and especially differential topology and singularity theory, the Eisenbud–Levine–Khimshiashvili signature formula gives a way of computing the Poincaré-Hopf index of a real, analytic vector field at an algebraically isolated singularity. It is named after David Eisenbud, Harold Levine, and George Khimshiashvili. Intuitively, the index of a vector field near a zero is the number of times the vector field wraps around the sphere. Because analytic vector fields have a rich algebraic structure, the techniques of commutative algebra can be brought to bear to compute their index. The signature formula expresses the index of an analytic vector field in terms of the signature of a certain quadratic form. == Nomenclature == Consider the ''n''-dimensional space R''n''. Assume that R''n'' has some fixed coordinate system, and write x for a point in R''n'', where Let ''X'' be a vector field on R''n''. For there exist functions such that one may express ''X'' as : To say that ''X'' is an ''analytic vector field'' means that each of the functions is an analytic function. One says that ''X'' is ''singular'' at a point p in R''n'' (or that p is a ''singular point'' of ''X'') if , i.e. ''X'' vanishes at p. In terms of the functions it means that for all . A singular point p of ''X'' is called ''isolated'' (or that p is an ''isolated singularity'' of ''X'') if and there exists an open neighbourhood , containing p, such that for all q in ''U'', different from p. An isolated singularity of ''X'' is called algebraically isolated if, when considered over the complex domain, it remains isolated. Since the Poincaré-Hopf index ''at a point'' is a purely local invariant (cf. Poincaré-Hopf theorem), one may restrict one's study to that of germs. Assume that each of the ƒ''k'' from above are ''function germs'', i.e. In turn, one may call ''X'' a ''vector field germ''. 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「Eisenbud–Levine–Khimshiashvili signature formula」の詳細全文を読む スポンサード リンク
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