Elementary symmetric polynomials and optimization

The mth elementary symmetric polynomial of degree n

$e_m(x_1, x_2, \ldots, x_n)$

is the sum of all terms containing a product of m variables. So, for example,

$\begin{align*} e_1(w, x, y, z) &= w + x + y + z \\ e_2(w, x, y, z) &= wx + wy + wz + xy + xz + yz \\ e_3(w, x, y, z) &= xyz + wyz + wxz + wxy \\ e_4(w, x, y, z) &= wxyz \end{align*}$
These polynomials came up in the previous post. The problem was choosing weights to minimize the variance of a weighted sum of random variables can be solved using elementary symmetric polynomials.