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 .