MATHEMATICA TUTORIAL

# 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,{x_{1},...,x_{n}}] | give the elementary symmetric polynomial in the variables |

SymmetricReduction[f,{x_{1},...,x_{n}}] | give a pair of polynomials in such that , where is the symmetric part and is the remainder |

SymmetricReduction[f,{x_{1},...,x_{n}},{s_{1},...,s_{n}}] | |

give the pair with the elementary symmetric polynomials in replaced by |

Functions for symmetric polynomial computations.

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This writes the polynomial in terms of elementary symmetric polynomials. The input polynomial is symmetric, so the remainder is zero.

In[2]:= |

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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.

In[3]:= |

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SymmetricReduction can be applied to polynomials with symbolic coefficients.

In[4]:= |

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