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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 Siegel's lemma ： ウィキペディア英語版 Siegel's lemma In transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue;〔 〕 Thue's proof used Dirichlet's box principle. Carl Ludwig Siegel published his lemma in 1929.〔, reprinted in Gesammelte Abhandlungen, volume 1; the lemma is stated on page 213〕 It is a pure existence theorem for a system of linear equations. Siegel's lemma has been refined in recent years to produce sharper bounds on the estimates given by the lemma.〔 and related numbers|journal = Journal für die reine und angewandte Mathematik|volume = 342|year = 1983|pages = 173–196}}〕 ==Statement== Suppose we are given a system of ''M'' linear equations in ''N'' unknowns such that ''N'' > ''M'', say :$a\_\; X\_1\; +\; \backslash cdots+\; a\_\; X\_N\; =\; 0$ :$\backslash cdots$ :$a\_\; X\_1\; +\backslash cdots+\; a\_\; X\_N\; =\; 0$ where the coefficients are rational integers, not all 0, and bounded by ''B''. The system then has a solution :$(X\_1,\; X\_2,\; \backslash dots,\; X\_N)$ with the ''X''s all rational integers, not all 0, and bounded by :$(NB)^.\backslash ,$〔 Lemma D.4.1, page 316.〕 gave the following sharper bound for the ''Xs: :$\backslash max|X\_j|\backslash le\; \backslash left(D^\backslash sqrt\backslash right)^$ where ''D'' is the greatest common divisor of the ''M'' by ''M'' minors of the matrix ''A'', and ''A''''T'' is its transpose. Their proof involved replacing the Dirichlet box principle by techniques from the geometry of numbers. 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「Siegel's lemma」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.