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.
| Out[1]= |  |
This writes the polynomial
in terms of elementary symmetric polynomials. The input polynomial is symmetric, so the remainder is zero.
| Out[2]= |  |
Here the elementary symmetric polynomials in the symmetric part are replaced with variables
. The polynomial is not symmetric, so the remainder is not zero.
| Out[3]= |  |
| Out[4]= |  |
