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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 Power sum symmetric polynomial ： ウィキペディア英語版 Power sum symmetric polynomial In mathematics, specifically in commutative algebra, the power sum symmetric polynomials are a type of basic building block for symmetric polynomials, in the sense that every symmetric polynomial with rational coefficients can be expressed as a sum and difference of products of power sum symmetric polynomials with rational coefficients. However, not every symmetric polynomial with integral coefficients is generated by integral combinations of products of power-sum polynomials: they are a generating set over the ''rationals,'' but not over the ''integers.'' ==Definition== The power sum symmetric polynomial of degree ''k'' in $n$ variables ''x''1, ..., ''x''''n'', written ''p''''k'' for ''k'' = 0, 1, 2, ..., is the sum of all ''k''th powers of the variables. Formally, :$p\_k\; (x\_1,\; x\_2,\; \backslash dots,x\_n)\; =\; \backslash sum\_^n\; x\_i^k\; \backslash ,\; .$ The first few of these polynomials are :$p\_0\; (x\_1,\; x\_2,\; \backslash dots,x\_n)\; =\; n\; ,$ :$p\_1\; (x\_1,\; x\_2,\; \backslash dots,x\_n)\; =\; x\_1\; +\; x\_2\; +\; \backslash cdots\; +\; x\_n\; \backslash ,\; ,$ :$p\_2\; (x\_1,\; x\_2,\; \backslash dots,x\_n)\; =\; x\_1^2\; +\; x\_2^2\; +\; \backslash cdots\; +\; x\_n^2\; \backslash ,\; ,$ :$p\_3\; (x\_1,\; x\_2,\; \backslash dots,x\_n)\; =\; x\_1^3\; +\; x\_2^3\; +\; \backslash cdots\; +\; x\_n^3\; \backslash ,\; .$ Thus, for each nonnegative integer $k$, there exists exactly one power sum symmetric polynomial of degree $k$ in $n$ variables. The polynomial ring formed by taking all integral linear combinations of products of the power sum symmetric polynomials is a commutative ring. 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「Power sum symmetric polynomial」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.