Sum of square of absolute values of roots of a polynomial

Question

If $\alpha_1,\dots,\alpha_n$ are roots of a polynomial $$P(z)=z^n+a_1z^{n-1}+\dots+a_{n-1}z+1,$$then how can one express the sum $$|\alpha_1|^2+\dots+|\alpha_n|^2$$in terms of $a_i$'s?

Thanks.

Answer 1

maybe this Lagrange's identity can usefull:

$$\sum_{1\le i<j\le n}|a_{i}-a_{j}|^2=n\sum_{i=1}^{n}|a_{i}|^2-|a_{1}+a_{2}+\cdots+a_{n}|^2$$ so we only find this closed form $$\sum_{1\le i<j\le n}|a_{i}-a_{j}|^2$$