# Symmetric Polynomials

A symmetric polynomial in variables is a polynomial that is invariant under arbitrary permutations of . Polynomials are called elementary symmetric polynomials in variables .

The fundamental theorem of symmetric polynomials says that every symmetric polynomial in can be represented as a polynomial in elementary symmetric polynomials in .

When the ordering of variables is fixed, an arbitrary polynomial can be uniquely represented as a sum of a symmetric polynomial , called the symmetric part of , and a remainder that does not contain descending monomials. A monomial is called descending iff .

 SymmetricPolynomial[k,{x1,…,xn}] give the elementary symmetric polynomial in the variables SymmetricReduction[f,{x1,…,xn}] give a pair of polynomials in such that , where is the symmetric part and is the remainder SymmetricReduction[f,{x1,…,xn},{s1,…,sn}] give the pair with the elementary symmetric polynomials in replaced by Functions for symmetric polynomial computations.

Here is the elementary symmetric polynomial of degree three in four variables:
 In:= Out= This writes the polynomial in terms of elementary symmetric polynomials. The input polynomial is symmetric, so the remainder is zero:
 In:= Out= Here the elementary symmetric polynomials in the symmetric part are replaced with variables . The polynomial is not symmetric, so the remainder is not zero:
 In:= Out= SymmetricReduction can be applied to polynomials with symbolic coefficients:
 In:= Out= 