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Concept
Elementary symmetric polynomial
Summary
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 positive integer d ≤ n, and it is formed by adding together all distinct products of d distinct variables.
The elementary symmetric polynomials in n variables X1, ..., Xn, written ek(X1, ..., Xn) for k = 1, ..., n, are defined by
and so forth, ending with
In general, for k ≥ 0 we define
so that ek(X1, ..., Xn) = 0 if k > n.
(Sometimes, 1 = e0(X1, ..., Xn) is included among the elementary symmetric polynomials, but excluding it allows generally simpler formulation of results and properties.)
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 non-increasing sequence of positive integers) λ = (λ1, ..., λm), one defines the symmetric polynomial eλ(X1, ..., Xn), also called an elementary symmetric polynomial, by
Sometimes the notation σk is used instead of ek.
The following lists the n elementary symmetric polynomials for the first four positive values of n.
For n = 1:
For n = 2:
For n = 3:
For n = 4:
The elementary symmetric polynomials appear when we expand a linear factorization of a monic polynomial: we have the identity
That is, when we substitute numerical values for the variables X1, X2, ...
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