Let's say I have a polynomial such as $$p(x) = x^4 + bx^3 + cx^2 + bx + 1.$$
I strongly suspect that, for any parameters, there are always two roots inside the unit circle and two roots outside of the unit circle.
What tools can I use to determine whether or not this is correct?
I am not necessarily looking for a solution to this problem but any answer that solves this problem would naturally contain tools that can be used in the general case.
Thank you.
On
As rogerl said, by symmetry the number of roots inside the unit circle is equal to the number outside. Now, how many are on the unit circle? Let's suppose $1$ is not a root, i.e. $2 + 2 b + c \ne 0$. The Möbius transformation $ w = i(1+z)/(1-z)$ ($z = (w-i)/(w+i)$) takes the unit circle (except for the point $1$) to the real line, and $p(z) = 0$ becomes $g(w) = (2b+c+2) w^4+(2c-12)w^2-2b+c+2 = 0$. The real solutions of $g(w) = 0$ correspond (in pairs, when nonzero) to nonnegative solutions of $(2b+c+2) t^2 + (2c-12) t - 2b + c + 2 = 0$. The roots of that quadratic are ${\dfrac {-c+6\pm 2\,\sqrt {{b}^{2}-4\,c+8}}{2\,b+c+2}}$. So for example when $b = 1$ and $c = 2$, the quadratic has two positive roots ($1$ and $1/3$), and all four of the roots of your quartic are on the unit circle.
$p\left(\frac{1}{x}\right) = \frac{1}{x^4}p(x)$. So unless there are roots on the unit circle (which is not ruled out in the problem as stated), there are two inside and two outside the unit circle.