Definition: A point $p$ on an affine plane curve $X$ defined by $f(z,w)=0$ is called a node of the plane curve if $p$ is a singular point of $X$, but the Hessian matrix of second partials is nonsingular at $p$.
In Rick Miranda's algebraic curve and Riemann surfaces, there is a factorization lemma for a affine plane curve at its node $p=(z_0,w_0)$:
But I can't see why this process can give us powers series $g$ and $h$ which are convergent near the node $p$. I think we need to carefully construct $g_i$ and $h_i$ because $f$ can be factorized formally into two divergent power series by this algorithm.