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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'' : 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'' :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
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