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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 Complete homogeneous symmetric polynomial ： ウィキペディア英語版 Complete homogeneous symmetric polynomial In mathematics, specifically in algebraic combinatorics and commutative algebra, the complete homogeneous symmetric polynomials are a specific kind of symmetric polynomials. Every symmetric polynomial can be expressed as a polynomial expression in complete homogeneous symmetric polynomials. ==Definition== The complete homogeneous symmetric polynomial of degree ''k'' in $n$ variables ''X''1, ..., ''X''''n'', written ''h''''k'' for ''k'' = 0, 1, 2, ..., is the sum of all monomials of total degree ''k'' in the variables. Formally, :$h\_k\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; \backslash sum\_\; X\_\; X\_\; \backslash cdots\; X\_.$ The formula can also be written as: :$h\_k\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =$ \sum_ X_^ X_^ \cdots X_^. Indeed, ''lp'' is just multiplicity of ''p'' in sequence ''ik''. The first few of these polynomials are :$h\_0\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; 1,$ :$h\_1\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; \backslash sum\_\; X\_j,$ :$h\_2\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; \backslash sum\_\; X\_j\; X\_k,$ :$h\_3\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; \backslash sum\_\; X\_j\; X\_k\; X\_l.$ Thus, for each nonnegative integer $k$, there exists exactly one complete homogeneous symmetric polynomial of degree $k$ in $n$ variables. Another way of rewriting the definition is to take summation over all sequences ''ik'', without condition of ordering $i\_p\; \backslash leq\; i\_$: :$h\_k\; (X\_1,\; X\_2,\; \backslash dots,X\_n)\; =\; \backslash sum\_$ \frac X_ X_ \cdots X_, here ''mp'' is the multiplicity of number ''p'' in the sequence ''ik''. For example :$h\_2\; (X\_1,\; X\_2)\; =\; \backslash fracX\_1^2\; +\backslash fracX\_1X\_2\; +\backslash fracX\_2X\_1\; +\; \backslash fracX\_2^2\; =\; X\_1^2+X\_1X\_2+X\_2^2.$ The polynomial ring formed by taking all integral linear combinations of products of the complete homogeneous 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「Complete homogeneous 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.