5
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.