# Wolfram Language & System 10.4 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# SymmetricReduction

SymmetricReduction[f,{x1,,xn}]
gives a pair of polynomials in such that , where is the symmetric part and is the remainder.

SymmetricReduction[f,{x1,,xn},{s1,,sn}]
gives the pair with the elementary symmetric polynomials in replaced by .

## DetailsDetails

• If is a symmetric polynomial, then is the unique polynomial in elementary symmetric polynomials equal to , and is zero.
• If is not a symmetric polynomial, then the output is not unique, but depends on the ordering of its variables.
• For a given ordering, a nonsymmetric polynomial can be expressed uniquely as a sum of its symmetric part and a remainder that does not contain descending monomials. A monomial is called descending if .
• Changing the ordering of the variables may produce different pairs .
• SymmetricReduction does not check to see that is a polynomial, and will attempt to symmetrize the polynomial part of .

## ExamplesExamplesopen allclose all

### Basic Examples  (3)Basic Examples  (3)

Write a symmetric polynomial as a sum of elementary symmetric polynomials:

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Write a nonsymmetric polynomial as a symmetric part and a remainder:

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Name the first two elementary symmetric polynomials and :

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