User:Loverthehater/sandbox

Examples
Symmetric polynomials in two variables x1, x2: $$x_1^3+ x_2^3-7$$ $$4 x_1^2x_2^2 +x_1^3x_2 + x_1x_2^3 +(x_1+x_2)^4$$ and in three variables x1, x2, x3: $$x_1 x_2 x_3 - 2 x_1 x_2 - 2 x_1 x_3 - 2 x_2 x_3 \,$$ 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 $$\prod_{1\leq i<j\leq n}(x_i-x_j)^2$$ 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 $$x_1 - x_2 \,$$ is not symmetric, since if one exchanges $$x_1$$ and $$x_2$$ one gets a different polynomial, $$x_2 - x_1$$. Similarly in three variables $$x_1^4x_2^2x_3 + x_1x_2^4x_3^2 + x_1^2x_2x_3^4$$ has only symmetry under cyclic permutations of the three variables, which is not sufficient to be a symmetric polynomial. However, the following is symmetric: $$x_1^4x_2^2x_3 + x_1x_2^4x_3^2 + x_1^2x_2x_3^4 + x_1^4x_2x_3^2 + x_1x_2^2x_3^4 + x_1^2x_2^4x_3$$