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?


  • $\begingroup$ Are the ($a_i$) real numbers or complex ones? Depending on the answer it might be easier to get a link, since a root will have its conjugate as a another root when ($a_i$) are real valued. $\endgroup$ – mvggz Dec 8 '14 at 9:31
  • $\begingroup$ They are complex. That's the main difficulty, I suppose. $\endgroup$ – Kunnysan Dec 8 '14 at 19:44

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 $$\sum_{1\le i<j\le n}|a_{i}-a_{j}|^2$$


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