Examples
In two variables X1, X2 one has symmetric polynomials like
and in three variables X1, X2, X3 one has for instance
There are many ways to make specific symmetric polynomials in any number of variables, see the various types below. An example of a somewhat different flavor is
where first a polynomial is constructed that changes sign under every exchange of variables, and taking the square renders it completely symmetric (if the variables represent the roots of a monic polynomial, this polynomial gives its discriminant).
On the other hand the polynomial in two variables
is not symmetric, since if one exchanges and one gets a different polynomial, . Similarly in three variables
has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial.
Read more about this topic: Symmetric Polynomial
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