Identifying when a family of palindromic polynomials has roots of unity

Question

I have the following family of polynomials

$$ (p-1)(1-x^{2(n+1)}) - x(1-x^{2n}) = 0 $$

where $p\in\mathbb{R}$. By construction this polynomial has roots on the unit circle, and this is reflected in the fact that it is anti-palindromic. I can divide out two roots, $x=1,-1$ and get the following polynomial

$$ (p-1)x^{2n} - x^{2n-1} + ... -x + (p-1) =0 $$ Which is a palindromic polynomial. What's interesting about these is that for every root $\alpha$ there is a corresponding root $\frac{1}{\alpha}$ and generically you can reduce this type of polynomial to something of the form

$$ x^m g(x+\frac{1}{x}) $$

for some value of $m$, where $g$ is a polynomial of lower order (for this case, $m=n$ and $g(x+\frac{1}{x}) = 2T_n (x + \frac{1}{x}) - \frac{2}{p-1}T_{n-1} (x+\frac{1}{x}) + ... + 1$ where $T_n$ is the $n$th Chebyshev polynomial of the first kind) which also has the information of the roots.

My question is how can I determine which, if any, roots of my original polynomial are roots of unity i.e. of the form $e^{i\frac{r}{s}\pi}, r,s\in\mathbb{Z}$, for general values of $p$.

There are special values of $p$ for which this equation is monic with all unital coefficients and the roots are all roots of unity, namely $p=0,1,2$. I expect that the roots depend continuously on the parameter $p$, with $p=1$ a singular point, but given that the polynomial is not monic for all other values of $p$ is it reasonable to expect that the roots are no longer simple roots of unity in these regions.

Answer 1

In [Uber die Anzahl der Wurzeln einer algebraischen Gleichung in einem Kreise, Math. Z., 14, 1922, 110-148], A. Cohn established the following result:

Let $p_n(z) = \sum_{k=0}^n a_k z^k$ be a polynomial of degree $n$ with complex coefficient, then all zeros of $p_n(z)$ lie on the unit circle if and only if $p_n(z)$ satisfies

(a) $a_{n-k} = \mu \bar{a}_k$, $0 \leq k \leq n$, $|\mu| = 1$;

(b) all the zeros of $p'_n(z)$ lie in or on the unit circle.

Edit You can find a lot of papers dealing with the same type of problem in the literature and such kind of polynomials are called self-reciprocal polynomials.