Apologies if the question is too elementary/something well-known.

I believe it is a well-known fact that the rational formal power series $F(z)=\frac{P(z)}{Q(z)}$ which have finite order under composition are (certain) Möbius transformations.

(By finite order I mean $F^{\langle n \rangle}(z) = z$ for some $n$, where the exponent denotes composition. Of course for this to be defined we should assume the power series has zero constant term.)

Is there some kind of classification of algebraic formal power series with finite order under composition?

**EDIT**: By "algebraic power series" I mean $P_n(z)F(z)^n + P_{n-1}(z)F(z)^{n-1} + \cdots + P_{0}(z) = 0$ for some $n\geq 1$ and polynomials $P_{n}(z), \ldots, P_0(z)$ (not all zero), where now the multiplication is the usual multiplication of power series, not composition: see e.g. Stanley's EC2, Definition 6.1.1. In certain contexts algebraic power series are the next natural step after rational power series.

**EDIT 2**: I guess a more precise way to formulate the question would be, say someone hands you an algebraic power series. Can you easily decide if it has finite order? One issue with this formulation is how do we “present” algebraic power series: with rational power series it is clear how we can compactly encode them, but with algebraic you need a little more than the minimal polynomial because there will be multiple roots in general.

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