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])[[1]]
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])[[1]],{x,0,3}]//Normal
Output 1:
$\frac{2 x^{5/3}}{9}-\frac{34 x^{7/3}}{81}+\frac{2 x^3}{3}-\sqrt[3]{x}-\frac{2 x}{3}$
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?
