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SymmetricReduction

SymmetricReduction[f, {x1, ..., xn}]
gives a pair of polynomials {p,q} in x_1,…,x_n such that f==p+q, where p is the symmetric part and q is the remainder.
SymmetricReduction[f, {x1, ..., xn}, {s1, ..., sn}]
gives the pair {p,q} with the elementary symmetric polynomials in p replaced by s_1,…,s_n.
  • If f is a symmetric polynomial, then p is the unique polynomial in elementary symmetric polynomials equal to f, and q is zero.
  • If f is not a symmetric polynomial, then the output p is not unique, but depends on the ordering of its variables.
  • For a given ordering, a nonsymmetric polynomial f can be expressed uniquely as a sum of its symmetric part p and a remainder q that does not contain descending monomials. A monomial is called descending if e_1≥…≥e_n.
  • Changing the ordering of the variables may produce different pairs {p,q}.
  • SymmetricReduction does not check to see that f is a polynomial, and will attempt to symmetrize the polynomial part of f.
EXAMPLESCLOSE ALL
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 s1 and s2:
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