# Series of implicit function (Puiseux expansion) - problem

This two one-line codes should represent the same thing, i.e. the first root of polynomial in $$y$$:

(y/.Solve[y^4+x y^3-x^2+x y==0,y])[]
Root[y^4+x y^3-x^2+x y/.y->#1&,1]


Then I computed identical series of the two expressions at $$x=0$$.

Input 1:

Series[(y/.Solve[y^4+x y^3-x^2+x y==0,y])[],{x,0,3}]//Normal


Output 1:

$$\frac{2 x^{5/3}}{9}-\frac{34 x^{7/3}}{81}+\frac{2 x^3}{3}-\sqrt{x}-\frac{2 x}{3}$$

Input 2:

Series[Root[y^4+x y^3-x^2+x y/.y->#1&,1],{x,0,3}]//Normal


Output 2:

-some error message-

But why I got right answer in the first case and error message in the second one?

Is this correct behavior?

What if I wanted to compute the same way series of $$y^5+x y^3-x^2+x y$$. Notice I changed the first term $$y^4$$ to $$y^5$$ and in this case output of "Solve" is in terms of "Root" because then radicals are impossible for 5th degree polynomial so I am unable to compute such series. How to do it also in this case? If I am not mistaken this series are called "Puiseux expansion". I can not find any other command in Matheamtica to compute such expansion for polynomials of degree bigger than $$4$$.

EDIT

I just noticed I got the same answer also in the second case but also with an error message because of which I overlooked the answer.

Anyway, is there a better method in Mathematica to compute such series - Puiseux expansion?

AsymptoticSolve[y^4 + x y^3 - x^2 + x y == 0, {y, 0}, {x, 0, 3}, Reals]

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