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
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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