Mathematical background

Let $P(x)$ be an univariate polynomial of degree $n$:

$$P(x)= a_0 x^n + a_{1} x^{n-1}+.....a_n=0$$

and $x_1,\ldots,x_n$ the real and complex roots of $P$. Let define $S_p$ as

$$S_p=\sum_{i =1}^{i =n} x_i^p$$

We have [4]:

$$\begin{eqnarray*} && a_0S_1+a_1=0\\ && a_0S_2+a_1S_1+2a_2=0\\ && \ldots\\ && ... ..._p=0\\ &&\ldots\\ && a_0S_n+a_1S_{n-1}+\ldots+a_{n-1}S_1+a_n=0 \end{eqnarray*}$$