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Symmetric Polynomials

A symmetric polynomial in variables x1, ..., xn is a polynomial that is invariant under arbitrary permutations of x1, ..., xn. Polynomials
are called elementary symmetric polynomials in variables x1, ..., xn.
The fundamental theorem of symmetric polynomials says that every symmetric polynomial in x1, ..., xn can be represented as a polynomial in elementary symmetric polynomials in x1, ..., xn.
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 is called descending iff e1≥...≥en.
SymmetricPolynomial[k,{x1,...,xn}]give the kth elementary symmetric polynomial in the variables x1, ..., xn
SymmetricReduction[f,{x1,...,xn}]give a pair of polynomials {p, q} in x1, ..., xn such that fp+q, where p is the symmetric part and q is the remainder
SymmetricReduction[f,{x1,...,xn},{s1,...,sn}]
give the pair {p, q} with the elementary symmetric polynomials in p replaced by s1, ..., sn

Functions for symmetric polynomial computations.

Here is the elementary symmetric polynomial of degree three in four variables.
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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.
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Here the elementary symmetric polynomials in the symmetric part are replaced with variables s1, s2, s3. The polynomial is not symmetric, so the remainder is not zero.
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SymmetricReduction can be applied to polynomials with symbolic coefficients.
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