Let $C$ be a curve in complex affine plane defined by an irreducible polynomial $f \in \mathbb{C}[x,y]$. If we create a Puiseux expansion of $C$ in the neighbourhhod of $(0,0)$, $C$ can be split into several branches (ie local parametrizations $(x,y)=(t^m,\sum_1^\infty a_r t^r)$).
My question is about tangents: is it possible for two different branches to have the same tangent at $(0,0)$? Can you give me an example of such irreducible $f$?
$f(x,y) = (x-y^2)(x+y^2) + x^3$.
More explanation : It's easy to check that it's irreducible. Now, for a polynomial $f = f_r + f_{r+1} + \dots $ where $f_d$ is the homogenous part of degree $d$, the branches of $f$ and $f_r$ are in bijection and have the same tangent direction as an easy calculation shows. In particular the branches of $f$ at $0$ are $x-y^2 = 0$ and $x+y^2 = 0$ which are indeed tangent.