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.
The elementary symmetric polynomials in n variables X1, …, Xn, written ek(X1, …, Xn) for k = 0, 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.
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 = (λ1, …, λm), one defines the symmetric polynomial eλ(X1, …, Xn), also called an elementary symmetric polynomial, by