Firstly, I know there are very similar questions (How to express a symmetric polynomial in terms of elementary symmetric polynomials and Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer? ) already. Unfortunately they didn't help me that much.
The task is to express some symmetric polynomials in terms of elementary symmetric polynomials. Which is always possible by a theorem which also says that if the initial polynomial is homogeneous of degree d, the resulting polynomial is isobaric of weight d.
For example $\sum_{i=1}^n x_i=S_1 $
$\sum_{i=1}^n x_i^2=S_1^2-2S_2$ because $\left(\sum_{i=1}^n x_i\right)^2=\sum_{i=1}^n x_i^2 + 2 \sum_{1\leq k < l \leq n}x_{k}x_l=S_1^2 +2S_2$
The case $\sum_{i=1}^n x_i^3$ is already significantly more complicated: One sees that $$S_1(S_1^2-2S_2)=\sum_{i=1}^n x_i^3+\sum_{r \neq s} x_{r} x_s^2$$ then $$S_1S_2=\sum_{i}x_i \cdot \sum_{r<s}x_r x_s=\sum_{r<s}x_r^2x_s+\sum_{r<s}x_r x_s^2+ \sum_{r<s \\ i \neq r,s}x_i x_r x_s=\sum_{r \neq s} x_{r} x_s^2+3S_3$$ Which gives us $\sum_{i=1}^n x_i^3=S_1^3-3S_1S_2+3S_3$
In the exercises it is asked to reshape other polynomials such as $\sum_{i \neq j } x_i^3 x_j$ and $\sum_{i=1}^n x_i^4$. The solutions of the exercises give the solutions after the first step, as if it was trivial. Personally I'm struggling a lot handling these expressions, am I missing something?
I would like to get some suggestions on how to learn solving these exercises. The theorem above gives some hints on how to choose the possible symmetric elementary polynomials, since they have to satisfy that relation. But how to find the coefficients in general?
Any suggestion is very much welcome.
