[Puiseux's theorem][1] asserts that

> given a polynomial equation $P(x,y)=0$, its solutions in $y$, viewed as functions of $x$, may be expanded as Puiseux series that are convergent in some neighbourhood of the origin ($0$ excluded, in the case of a solution that tends to infinity at the origin). In other words, every branch of an algebraic curve may be locally (in terms of $x$) described by a Puiseux series.

I'm also interested in expansions around $x=\infty$, which can be obtained by applying Puiseux's theorem to the equivalent polynomial equation $$t^dP(t^{-1},y)=0,$$
with $t=x^{-1}$ and $d =\deg_x P(x,y).$

**My questions are as follows:**

 Let $k_0,n$ be integers, and suppose we have a relationship of the form
$$y=\sum_{k=k_0}^\infty c_k \frac{1}{x^{k/n}} $$
with complex $\{c_k\}_k$, and $x$ sufficiently large. Is there a polynomial curve $P(x,y)=0$, with one of the branches of which having a Puiseux series expansion as in above? 

A different version of this question is an asymptotic one: Suppose
$$y \sim \sum_{k=k_0}^\infty c_k \frac{1}{x^{k/n}}, \; \text{as }x \to \infty. $$
Is there a polynomial curve $P(x,y)=0$, such that one of its branches admits an asymptotic Puiseux series expansion as above?


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In the case where the answer to both of my questions is 'no', I'm still interested in any kind of information/reference regarding analytic curves $f(x,y)=0$ with branches which can be expanded in a Puiseux series as $x \to \infty$ (possibly in the asymptotic sense).

If the answer to any of my questions is a 'yes', I'd like to know: how to start constructing such a polynomial curve? does the number $n$ in the Puiseux series say anything about the degree of $P(x,y)$?

Thank you!  


[1]: https://en.wikipedia.org/wiki/Puiseux_series