gives a pair of polynomials in such that , where is the symmetric part and is the remainder.
gives the pair with the elementary symmetric polynomials in replaced by .
- 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 .