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 ^(th) 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.
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This writes the polynomial 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 . 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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