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I have one polynomial $Q(z) = \sum\limits_{n=0}^{2a-1} z^n c_n$, with $c_n \in \mathbb{R}$ and $c_{2a-1}\neq0$. Using Rouché's Theorem, I could locate them as inside or outside the unit disk, with $a$ inner roots and $a-1$ outer roots.

Then, running some simulations, I noticed that the products of the inner roots and, obviously, the outer ones too, were real (despite the complex nature of them individually).

So, now I need those values, and to be sure they are real would help a lot. So my question is: are there any theorem that shows for some polynomials the products of its inner and outer roots are real?

Thanks,

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Since all the coefficients of $Q$ are real, non real roots come in pairs, $\alpha$, $\overline \alpha$, and since $\alpha\overline\alpha = |\alpha|^2$ is real, the product of the roots that satsify $|\alpha| < 1$ are real.

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  • $\begingroup$ The real point is that $|\alpha| < 1$ iff $| \overline \alpha| < 1$. $\endgroup$
    – lhf
    Jun 10 '15 at 13:58

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