composition of Puiseux series?

Question

Formal series over a field (or ring) k can be composed in the following sense: given $y \in k[[x]]$, $y$ lying in the maximal ideal, there exists a unique map of topological rings $k[[x]] \to k[[x]]$ mapping $x$ to $y$.

Is there a similar statement for Puiseux series, $\cup_n k((x^{1/n}))$? I am not sure what a correct statament should sound like, but at least the composition of Puiseux series should specialise to the operation mentioned abouve for formal series.

My intuition is that formal series are germs of analytic functions and their composotion is the compostion of function on the formal level. Puiseux series are germs of multi-valued functions around a branching point and thus also ought to have some operation corresponding to composition defined on them.

Answer 1

Yes, if the second series is in the maximal ideal of $k[[x^{1/n}]]$ for some $n$. Simply write it as $x^{a/n}$ times a constant times a series whose leading term is $1$. Then we can take the $m$th root of this series for any integer $m$, with the $m$th root of $x^{a/n}$ equal to $x^{a/(mn)}$, the $m$th root of the constant as usual, and the $m$th root of $1$ plus higher order terms defined by composing the higher-order terms with the formal power series for $\sqrt[m]{1+x}$.

Once we take the $m$th root, we can compose in the normal way with any formal power series in $k[[x^{1/m}]]$ or formal Laurent series in $k((x^{1/m}))$ to get a good notion of composition for Puisex series.