Elementary symmetric polynomial

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 $d \leq n$ , and it is formed by adding together all distinct products of $d$ distinct variables.

The elementary symmetric polynomials in $n$ variables $X 1, \dots, X n$ , written $e k (X 1, \dots, X n)$ for $k = 0, 1, \dots, n$ , are defined by

and so forth, ending with

In general, for $k \geq 0$ we define

so that $e k (X 1, \dots, X n) = 0$ if $k > n$ .

Thus, for each positive integer $k$ less than or equal to $n$ there exists exactly one elementary symmetric polynomial of degree $k$ in $n$ variables. To form the one that has degree $k$ , we take the sum of all products of $k$ -subsets of the $n$ 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) $λ = (λ 1, \dots, λ m)$ , one defines the symmetric polynomial $e λ (X 1, \dots, X n)$ , also called an elementary symmetric polynomial, by

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