I am asking this question mainly to probe the knowledge of people already familiar with this problem, otherwise I would advise caution to the unfamiliar trying to use computation, this can be really exhausting and frankly disappointing, hence my question, essentially made for people already acquainted with this problem. References, if any, would be greatly appreciated!
Let $R(t)$ be a rational function, like a polynomial $R(t)=t^2-t+1$ or a fraction of polynomials $R(t)=\frac{t^2+t+1}{t^3+2}$, etc.
Given $n$ (usually $n=2$), how do I know whether there exists $f$ such that
$f \circ f \circ f \cdots \circ f(t) = R(t)$ where the numbers of $\circ$ is $n-1$, and where $f$ is a power series with indices in $\mathbb{Z}$, i.e. $f$ has the form
$f(t)=\sum_{i=-\infty}^{+\infty}a_it^i$
and is there a nice algorithm to compute $f$?
To make things simpler, I am asking mainly about the algebraic existence (no topology), but it would be nice if the $f$'s could converge at least in some specific interval...
In any case, if $n=2$, can the rational functions obtained this way be characterized? If $n$ is any number $\geq 2$, can I get all the rational functions?
The reason I am asking this is that I've been through extremely tedious calculations to find solutions to recreational mathematics functional equations, usually like $f(f(x))=P(x)$, and while sometimes I was able to find such an $f$, sometimes I wasn't, for instance I think I spent way too much time trying to find $f$ satisfying $f(f(x))=x^2-x+1$. I don't know whether there exists a solution to this problem (it seems like it does not), the original problem was much simpler and did not require finding $f$.
