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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 Rabinowitsch trick ： ウィキペディア英語版 Rabinowitsch trick In mathematics, the Rabinowitsch trick, introduced by George Yuri Rainich and published under the pseudonym , is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called ''weak'' Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows. Let ''K'' be an algebraically closed field. Suppose the polynomial ''f'' in ''K''() vanishes whenever all polynomials ''f''1,....,''f''''m'' vanish. Then the polynomials ''f''1,....,''f''''m'', 1 − ''x''0''f'' have no common zeros (where we have introduced a new variable ''x''0), so by the weak Nullstellensatz for ''K''() they generate the unit ideal of ''K''(). Spelt out, this means there are polynomials $g\_0,g\_1,\backslash dots,g\_m\; \backslash in\; K()$ such that : $1\; =\; g\_0(x\_0,x\_1,\backslash dots,x\_n)\; (1\; -\; x\_0\; f(x\_1,\backslash dots,x\_n))\; +\; \backslash sum\_^m\; g\_i(x\_0,x\_1,\backslash dots,x\_n)\; f\_i(x\_1,\backslash dots,x\_n)$ as an equality of elements of the polynomial ring $K()$. Since $x\_0,x\_1,\backslash dots,x\_n$ are free variables, this equality continues to hold if expressions are substituted for some of the variables; in particular, it follows from substituting $x\_0\; =\; 1/f(x\_1,\backslash dots,x\_n)$ that : $1\; =\; \backslash sum\_^m\; g\_i(1/f(x\_1,\backslash dots,x\_n),x\_1,\backslash dots,x\_n)\; f\_i(x\_1,\backslash dots,x\_n)$ as elements of the field of rational functions $K(x\_1,\backslash dots,x\_n)$, the field of fractions of the polynomial ring $K()$. Moreover, the only expressions that occur in the denominators of the right hand side are ''f'' and powers of ''f'', so rewriting that right hand side to have a common denominator results in an equality on the form : $1\; =\; \backslash frac$ for some natural number ''r'' and polynomials $h\_1,\backslash dots,h\_m\; \backslash in\; K()$. Hence : $f(x\_1,\backslash dots,x\_n)^r\; =\; \backslash sum\_^m\; h\_i(x\_1,\backslash dots,x\_n)\; f\_i(x\_1,\backslash dots,x\_n)$, which literally states that $f^r$ lies in the ideal generated by ''f''1,....,''f''''m''. This is the full version of the Nullstellensatz for ''K''(). ==References== * * 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「Rabinowitsch trick」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.