Let $f(x,y)\in \mathbf{C}[x,y]$ a plane curve $f(0,0)=0$. It's well knowing that using a Newton's method, we can find a Puiseux series associated with each branch of the curve, i.e., we have an analytic parametrization $g_i$ of each branch: Here for some natural number $n$, each $g_i \in \mathbf{C}[x^{\frac{1}{n}}]$ with $g_i(0) = 0$. Puiseux proved that each $g_i$ converges in a neighborhood of the origin.
When the parametrization $g_i$ is finite (it is not an infinite series) then we can prove that $g_i$ converges absolutely. Is it true in the infinite case? does g_i converges absolutely in the general case?.
