Conjecture on the number of roots of $z^n + P(z)$ within the unit disk

Question

Some other people and I have noticed that the following seems to be true.

Fix an integer polynomial $P \in \mathbb{Z}[x]$. Let $a_n$ be the number of roots of $z^n + P(z) = 0$ that lie in the unit disk, ie $|z| \le 1$. Then there exists positive integer $k$ and nonnegative integer $m$ such that $a_{n+k} = a_n + m$ holds for all sufficiently large $n$.

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Here's some graphs of small examples. $k$ here is the suspected value.

• $P(x) = x + 1, k = 3$
• $P(x) = x^{11} + 1, k = 33$
• $P(x) = x^{3} + x + 1, k = 12$
• $P(x) = x^{2} + x + 2, k = 93$

As such, the question we have is whether this conjecture holds. If it does not, then the followup question becomes why it seems to be periodic for small $n$, as well as for what, if any, subset of $\mathbb{Z}[x]$ it holds. If it is does end up being true, then we wonder if there's any nice relation between $P$ and $k, m$ (though $93$ being the period suggests not).

Answer 1

I am afraid that this is too good to be true. Consider the polynomial $P(z)=2z+2$. There are two points on the unit circle, $\alpha$ and $\bar{\alpha}$, for which $|P(\alpha)|=|P(\bar{\alpha})|=1$. Also note that $\alpha$ is not a root of unity (it solves the equation $(2z+2)(2/z+2)=1$, i.e., $4(z+1/z)=-7$, so, it is not even an algebraic integer).

$\alpha$ and $\bar{\alpha}$ partition the unit circle onto two arcs $D_+$ and $D_-$, on $D_+$ we have $|P(z)|>1$, on $D_-$ we have $|P(z)|<1$. Let's count the number of roots of $z^n+P(z)$ inside the unit circle by the argument principle. On $D_+$ we have $z^n+P(z)=P(z)(1+z^n/P(z))$, the second factor lies in the right half-plane, thus, its argument is changed by $O(1)$ when we go along $D_+$. The same holds for the argument change of $P(z)$. On $D_-$, we write $z^n+P(z)=z^n(1+P(z)/z^n)$, again the second factor has bounded change of the argument, while the first factor has the change of argument $\theta\cdot n$, where $\theta$ is the length of $D_-$. But $\theta/\pi$ is irrational (that is equivalent to $\alpha$ not being a root of unity), thus, $\lim a_n/n$ is irrational. This contradicts to the existence of such $k$ and $m$.