|
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symmetric polynomials. That is, any symmetric polynomial ''P'' is given by an expression involving only additions and multiplication of constants and elementary symmetric polynomials. There is one elementary symmetric polynomial of degree ''d'' in ''n'' variables for each nonnegative integer ''d'' ≤ ''n'', and it is formed by adding together all distinct products of ''d'' distinct variables. ==Definition== The elementary symmetric polynomials in variables ''X''1, …, ''X''''n'', written ''e''''k''(''X''1, …, ''X''''n'') for ''k'' = 0, 1, ..., ''n'', are defined by : and so forth, ending with :. In general, for ''k'' ≥ 0 we define : so that if . Thus, for each positive integer less than or equal to there exists exactly one elementary symmetric polynomial of degree in variables. To form the one that has degree , we take the sum of all products of -subsets of the variables. (By contrast, if one performs the same operation using ''multisets'' of variables, that is, taking variables with repetition, one arrives at the complete homogeneous symmetric polynomials.) Given an integer partition (that is, a finite decreasing sequence of positive integers) λ = (λ1, …, λ''m''), one defines the symmetric polynomial , also called an elementary symmetric polynomial, by : . Sometimes the notation σ''k'' is used instead of ''e''''k''. 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「elementary symmetric polynomial」の詳細全文を読む スポンサード リンク
|