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.
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sorry to ask such a question, but what is an elementary symmetric polynomial? in this case the polynomial is a function of what? – Math-fun Oct 20 '15 at 12:06
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In this case the elementary symmetric polynomials would be $s_1 = \alpha + \beta + \gamma + \delta$, $s_2=\alpha\beta + \alpha\gamma + \alpha\delta + \beta\gamma + \beta\delta + \gamma\delta$, $s_3= \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta$ and $s_4= \alpha\beta\gamma\delta$. – guta1 Oct 20 '15 at 12:18
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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.
Ivan Neretin
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For a general approach, check: http://math.stackexchange.com/questions/14051/symmetric-polynomials-and-the-newton-identities/14061#14061 – Macavity Oct 20 '15 at 13:15
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