Puiseux's theorem 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?
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!
