Let $Q$ a polynomial in $\mathbb{C}[X]$ (or in $\mathbb{K}[X]$ with $\mathbb{K}$ a field of characteristic zero).
I wonder what is the commutant of $Q$, i.e. the set of polynomials $P$ such that $P \circ Q = Q \circ P$, where $\circ$ is the composition. More precisely, is there any works, any results on that topic ?
All the followings remarks also works in $\mathbb{K}[X]$, where the (formal) Bottcher coordinates are still defined. This holds because the characteristic of $\mathbb{K}$ is zero.
If $Q$ is constant or of degree $1$, the answer is easy. I suppose that $Q$ is of degree $d_0$, and $d_0$ greater than $1$. Conjugating $Q$ by an homothety, I suppose that the leading coefficient of $Q$ is $1$.
The constants polynomials commuting with $Q$ are its fixed points.
Let $d$ a natural integer. Using the Bottcher coordinates $\Phi$ at infinity, it is possible to show that for all integer $d$ there are exactly $d_0-1$ Laurent séries $L$ in $1/X$ of degree $d$ (i.e. $L = \sum_{k = - \infty}^d a_k X^k$ with $a_d \neq 0$) commuting with $Q$. Indeed, in these coordinates $Q$ is $X^{d_0}$, and $L$ must be $\zeta X^d$ where $\zeta^{d_0-1} = 1$. This gives the explicit formula $L(X) = \Phi^{-1}( \zeta \Phi(X)^d )$.
But it is not easy at all to caracterize when $L$ is a polynomial in X.
For instance, for the polynomial $Q = X^2 + 1$, I am tempted to say that a polynomial $P$ of degree greater than 0 commutes with $Q$ if and only if $P$ is of the form $P = Q\circ ... \circ Q$. The preceeding paragraphs shows that it is true if the degree of $P$ is a power of two, but it remains to show that $L(X) = \Phi^{-1}( \Phi(X)^d )$ is not a polynomial when $d$ is not a power of two.
Is it possible to calculate the commutant of $Q$ in this case ? Or in some simple cases where $Q$ is not conjugate to a Tchebychev polynomial or a power of $X$ ?
I've found a detailed caracterisation of commutation for polynomials in $\mathbb{C}[X]$. See the following article of Ritt : https://www.jstor.org/stable/1989297
The theorem says that two polynomials $P$ and $Q$ of degree greater than 1 commute if and only if, up to conjugacy in $\mathbb{C}[X]$ :
The example of $Q = X^2+1$ belongs to the third case with r=1, thus the commutant of $Q$ is formed by its iterates and the constant polynoms where the constant is a fixed point of $Q$.