Expressing in terms of symmetric polynomials. How to express $$a^7+b^7+c^7$$  in terms of symmetric polynomials ${\sigma}_{1}=a+b+c$, ${\sigma}_{2}=ab+bc+ca$ and ${\sigma}_{3}=abc$ ?

We can take a slightly different approach. We have that $a,b,c$ are the roots of the polynomial:
$$ p(x)=x^3-\sigma_1 x^2+\sigma_2 x-\sigma_3 $$
hence the eigenvalues of the companion matrix:
$$ M = \left(\begin{array}{ccc} 0 & 0 & \sigma_3 \\ 1 & 0 & -\sigma_2 \\ 0 & 1 & \sigma_1\end{array}\right) $$
so:
$$\begin{eqnarray*} a^7+b^7+c^7 &=& \operatorname{Tr}\left(M^7\right)\\&=&\sigma_1^7 - 7 \sigma_1^5 \sigma_2 + 14 \sigma_1^3 \sigma_2^2 - 7 \sigma_1 \sigma_2^3 + 7 \sigma_1^4 \sigma_3 - 21 \sigma_1^2 \sigma_2 \sigma_3 + 
 7 \sigma_2^2 \sigma_3 + 7 \sigma_1 \sigma_3^2.\end{eqnarray*} $$
