$(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials?

Question

Is it possible to express $(\alpha +\beta - \gamma - \delta)(\alpha -\beta + \gamma - \delta)(\alpha -\beta - \gamma + \delta)$ in terms of elementary symmetric polynomials ? What I tried was rewriting each factor as a sum of two terms, e.g. $(\alpha +\beta - \gamma - \delta) = (\alpha - \gamma) + (\beta - \delta)$, but that didn't lead anywhere.

Answer 1

Well, it is symmetric, so this must be possible. Indeed, your polynomial is $s_1^3-4s_1s_2+8s_3$.

Upd. How did I find this? Well, it is of degree 3, so it must be some combination of $s_1^3,\;s_1s_2$, and $s_3$. Now look at the term $\alpha^3$ (also $\beta^3$, etc.) - it must have come from $s_1^3$, so we subtract that. Then look at the terms $\alpha^2\beta$ and the like. These are from $s_1s_2$, and they go with a coefficient -4, so we subtract $-4s_1s_2$, and instantly recognize the rest.