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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 Puiseux series ： ウィキペディア英語版 Puiseux series In mathematics, Puiseux series are a generalization of power series, first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850,〔Puiseux (1850, 1851)〕 that allows for negative and fractional exponents of the indeterminate ''T''. A Puiseux series in the indeterminate ''T'' is a Laurent series in ''T''1/''n'', where ''n'' is a positive integer. A Puiseux series may be written as: :$\backslash sum\_^\; a\_iT^,$ where is an integer and is a positive integer. Puiseux's theorem, sometimes also called Newton–Puiseux theorem, asserts that, given a polynomial equation $P(x,y)=0$, its solutions in , viewed as functions of , may be expanded as Puiseux series that are convergent in some neighbourhood of the origin (0 excluded, in the case of a solution that tends to infinity at the origin). In other words, every branch of an algebraic curve may be locally (in terms of ) described by a Puiseux series. The set of Puiseux series over an algebraically closed field of characteristic 0 is itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of Laurent series. This statement is also referred to as Puiseux's theorem, being an expression of the original Puiseux theorem in modern abstract language. == Field of Puiseux series == If ''K'' is a field then we can define the field of Puiseux series with coefficients in ''K'' (or ''over'' ''K'') informally as the set of formal expressions of the form :$f\; =\; \backslash sum\_^\; c\_k\; T^$ where ''n'' and $k\_0$ are a nonzero natural number and an integer respectively (which are part of the datum of ''f''): in other words, Puiseux series differ from formal Laurent series in that they allow for fractional exponents of the indeterminate as long as these fractional exponents have bounded denominator (here ''n''), and just as Laurent series, Puiseux series allow for negative exponents of the indeterminate as long as these negative exponents are bounded (here by $k\_0$). Addition and multiplication are as expected: one might define them by first “upgrading” the denominator of the exponents to some common denominator and then performing the operation in the corresponding field of formal Laurent series. In other words, the field of Puiseux series with coefficients in ''K'' is the union of the fields $K(\backslash !(T^)\backslash !)$ (where ''n'' ranges over the nonzero natural numbers), where each element of the union is a field of formal Laurent series over $T^$ (considered as an indeterminate), and where each such field is considered as a subfield of the ones with larger ''n'' by rewriting the fractional exponents to use a larger denominator (e.g., $T^$ is identified with $T^$ as expected). This yields a formal definition of the field of Puiseux series: it is the direct limit of the direct system, indexed over the non-zero natural numbers ''n'' ordered by divisibility, whose objects are all $K(\backslash !(T)\backslash !)$ (the field of formal Laurent series, which we rewrite as :$K(\backslash !(T\_n)\backslash !)$ for clarity), with a morphism :$K(\backslash !(T\_m)\backslash !)\; \backslash to\; K(\backslash !(T\_n)\backslash !)$ being given, whenever ''m'' divides ''n'', by $T\_m\; \backslash mapsto\; (T\_n)^$. 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「Puiseux series」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.