# Polynom as sum/product of symmetric polynoms

I have a polynom $(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1)$ and I need to express as sum/product of elemental symmetric polynoms $s_1,s_2,s_3$. I know there is an algoritm for that, but can't understand how it works. Can somebody explain it to me?

• You usually can deal with these exercises without the algorithm. Just expand the expression and think of how to group the monomials nicely. If you use the algorithm you may have to spend a great deal of time. It works really slowly. It's mainly used to show that every symmetric polynomial could be represented with the elemental symmetric polynomials. – brick Nov 19 '14 at 20:16
• I really doubt I can guess answer like this. – Dark Archon Nov 19 '14 at 20:33

## 1 Answer

$(x_1^2x_3 + x_2^2x_1 + x_3^2x_2)(x_1^2x_2 + x_2^2x_3 + x_3^2x_1) = (x_1^4x_2x_3 + x_2^4x_1x_2 + x_3^4x_1x_2) + 3(x_1x_2x_3)^2 + ((x_1x_2)^3 + (x_2x_3)^3+(x_1x_3)^3).$

Let's deal first with $(x_1x_2)^3 + (x_2x_3)^3+(x_1x_3)^3 = A$.

$s_2^2 = (x_1x_2)^2 + (x_2x_3)^2 + (x_1x_3)^2 + 2(x_1^2x_2x_3 + x_1x_2^2x_3 + x_1x_2x_3^2) = (x_1x_2)^2 + (x_2x_3)^2 + (x_1x_3)^2 + 2s_3s_1$

So $(x_1x_2)^2 + (x_2x_3)^2 + (x_1x_3)^2 = s_2^2 - 2s_1s_3$.

Now $s_2(s_2^2 - 2s_1s_3) = (x_1x_2 + x_2x_3 + x_1x_3)((x_1x_2)^2 + (x_2x_3)^2 + (x_1x_3)^2) = (x_1x_2)^3 + (x_2x_3)^3 + (x_1x_3)^3 + x_1x_2^2x_3^3 + x_1x_2^3x_3^2 + x_1^2x_2x_3^3 + x_1^2x_2^3x_3 + x_1^3x_2x_3^2 + x_1^3x_2^2x_3 = A + s_3(x_1x_2^2 + x_1^2x_2 + x_2x_3^2 + x_2^2x_3 + x_1x_3^2 + x_1^2x_3).$

The only thing left is to calculate $x_1x_2^2 + x_1^2x_2 + x_2x_3^2 + x_2^2x_3 + x_1x_3^2 + x_1^2x_3.$

$x_1x_2^2 + x_1^2x_2 + x_2x_3^2 + x_2^2x_3 + x_1x_3^2 + x_1^2x_3 = (x_1x_2^2 + x_1^2x_2) + (x_2x_3^2 + x_2^2x_3) + (x_1x_3^2 + x_1^2x_3) = x_1x_2(x_1 + x_2) + x_2x_3(x_2 + x_3) + x_1x_3(x_1 + x_3) = x_1x_2(s_1-x_3) + x_2x_3(s_1 - x_1) + x_1x_3(s_1 - x_2) = s_1s_2 - 3s_3.$

And finally $s_2(s_2^2 - 2s_1s_3) = A + s_3(s_1s_2 - 3s_3)$, so $A = s_2(s_2^2 - 2s_1s_3) - s_3(s_1s_2 - 3s_3).$

$x_1^4x_2x_3 + x_2^4x_1x_2 + x_3^4x_1x_2 = s_3(x_1^3 + x_2^3 + x_3^3)$ shouldn't be more difficult. Try it yourself!