Why are these sets divisible by n?

Question

Suppose we have a polynomial $z \to f_c(z)$ defined over $\mathbb Z$ with a free parameter $c$, for instance $z \to z^2 + c$ and we consider the iterates $z \to f_c^{(n)}(z)$ and define the polynomials $g_n(c) = f_c^{(n)}(0)$. That is, the roots of $g_n$ correspond to those $c$ for which $0$ has a period of size dividing $n$. For example, with $f_c(z) = z^2 + c$: \begin{gather*} g_2(c) = c^2 + c \\ g_3(c) = (c^2 + c)^2 + c \end{gather*} and so on. We also define $h_n(x) \mid g_n(x)$ to be the polynomial with roots corresponding to $c$ so that $0$ has period exactly $n$. Then: \begin{gather*} h_2(c) = c+1 \\ h_3(c) = c^3 + 2c^2 + c + 1 \end{gather*} and so on.

In particular, if $f(z)$ has degree $d$, then $g_n(z)$ has degree $d^{n-1}$ and by möbius inversion: $$\deg(h_n) = \sum_{m\mid n}\mu\left(\frac{n}{m}\right)d^{m-1}.$$

Now it turns out to be true that we have the following congruence: $$\sum_{m\mid n}\mu\left(\frac{n}{m}\right)d^m \equiv 0 \pmod{n}$$ and therefore, at least when $\gcd(d,n) = 1$, we have that $\deg(h_n)$ is divisible by $n$. Is there a natural way to partition the roots of $h_n$ (corresponding to $c$ so that $0$ has an orbit of period exactly $n$) into sets of size $n$ (or perhaps $n$ sets?)?

I don't see a straightforward way of doing this, especially because $h_3(c) = c^3 + 2c^2 + c + 1$ in the example above turns out not to generate a Galois extension and so we cannot write all 3 roots as algebraic expressions in one of the roots.

(I believe it is a conjecture that the $h_n$ are irreducible over $\mathbb Q$ at least for $f(z) = z^2 + c$. I would be very happy if someone could provide a reference for this conjecture and also perhaps what we expect to happen in general.)

Answer 1

As Asvin mentioned, the polynomials that you're looking at are often called Gleason polynomials, although sometimes people only require that the critical point 0 have finite orbit, rather that being periodic. (Sometimes those preperiodic versions are called "Misiurewicz polynomials".) I'm not sure who first asked (conjectured) that the Gleason polynomials are irreducible, but it was listed as a problem worth studying at an AIM workshop on Postcritically finite maps in complex and arithmetic dynamics:

http://aimpl.org/finitedynamics/3/

As for what's true in general, it depends on what you mean. Do you want to restrict to 1-parameter families of polynomials, in which case you'll get analogous Gleason-polynomials that are polynomials in the parameter. Or you could take higher dimensional families and get multi-variable Gleason polynomials. All of these lead to interesting, and mostly open, questions. For recent preprints on irreduciblity of some Misiurewicz polynomials in a family of rational maps that aren't polynomials, see:

• Misiurewicz polynomials for rational maps with nontrivial automorphisms
• Misiurewicz polynomials for rational maps with nontrivial automorphisms II

Answer 2

In the case of $f_c(z) = z^2 + c$, the recent preprint Buff, Floyd, Koch, and Parry - Factoring Gleason polynomials mod 2 shows that we can factor $g_n(c) \pmod 2$ into irreducibles of degree $n$ in the odd case and something similar works for $n$ even too. In particular, see Theorem 1.2 and Theorem 1.6.

Unfortunately, the techniques don't seem to generalize in a straightforward manner for arbitrary $f_c(z)$. They do seem to handle the case of $f_c(z)= z^{p^n} + c$ for $p$ a prime however.