Symmetric Polynomials

A symmetric polynomial in variables (Subscript[x, 1]),…,(Subscript[x, n]) is a polynomial that is invariant under arbitrary permutations of (Subscript[x, 1]),…,(Subscript[x, n]). Polynomials

are called elementary symmetric polynomials in variables (Subscript[x, 1]),…,(Subscript[x, n]).

The fundamental theorem of symmetric polynomials says that every symmetric polynomial in (Subscript[x, 1]),…,(Subscript[x, n]) can be represented as a polynomial in elementary symmetric polynomials in (Subscript[x, 1]),…,(Subscript[x, n]).

When the ordering of variables is fixed, an arbitrary polynomial f can be uniquely represented as a sum of a symmetric polynomial p, called the symmetric part of f, and a remainder q that does not contain descending monomials. A monomial c(Subscript[x, 1]^(Subscript[e, 1]))… (Subscript[x, n]^(Subscript[e, n])) is called descending iff (Subscript[e, 1])≥…≥(Subscript[e, n]).

Functions for symmetric polynomial computations.

Here is the elementary symmetric polynomial of degree three in four variables:

SymmetricPolynomial[3,{x,y,z,t}]

This writes the polynomial (((x+y))^2)+(((x+z))^2)+(((z+y))^2) in terms of elementary symmetric polynomials. The input polynomial is symmetric, so the remainder is zero:

SymmetricReduction[(((x+y))^2)+(((x+z))^2)+(((z+y))^2),{x,y,z}]

Here the elementary symmetric polynomials in the symmetric part are replaced with variables (Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3]). The polynomial is not symmetric, so the remainder is not zero:

SymmetricReduction[(x^5)+(y^5)+(z^4),{x,y,z},{(Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3])}]

(RefLink[SymmetricReduction, paclet:ref/SymmetricReduction]) can be applied to polynomials with symbolic coefficients:

SymmetricReduction[(x^5)+(y^5)+(z^4)+a (x^4)+b (y^4)+c (z^5),{x,y,z},{(Subscript[s, 1]),(Subscript[s, 2]),(Subscript[s, 3])}]