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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 Artin approximation theorem ： ウィキペディア英語版 Artin approximation theorem In mathematics, the Artin approximation theorem is a fundamental result of in deformation theory which implies that formal power series with coefficients in a field ''k'' are well-approximated by the algebraic functions on ''k''. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = C); and an algebraic version of this theorem in 1969. ==Statement of the theorem== Let :x = ''x''1, …, ''x''''n'' denote a collection of ''n'' indeterminates, ''k''x the ring of formal power series with indeterminates x over a field ''k'', and : y = ''y''1, …, ''y''''m'' a different set of indeterminates. Let :''f''(x, y) = 0 be a system of polynomial equations in ''k''(y ), and ''c'' a positive integer. Then given a formal power series solution ŷ(x) ∈ ''k''x there is an algebraic solution y(x) consisting of algebraic functions (more precisely, algebraic power series) such that :ŷ(x) ≡ y(x) mod (x)''c''. 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「Artin approximation theorem」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.