Symmetric polynomials

A symmetric polynomial is a polynomial in several variables that does not change when any two of its variables are swapped. For example, if $f(x, y) = x^2 + y^2$, then $f(x, y) = f(y, x)$, so it is symmetric. In general, a polynomial $P(x_1, x_2, ..., x_n)$ is symmetric if swapping any $x_i$ and $x_j$ leaves $P$ unchanged.

Important properties

• Symmetric polynomials remain the same under any permutation of their variables.

• Elementary symmetric polynomials are basic building blocks: for $n$ variables, these are sums of products of variables taken $k$ at a time.

• Any symmetric polynomial can be written as a polynomial in the elementary symmetric polynomials.