Let F ∈ K[X, Y ] be a polynomial of total degree D defined over a field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all rational Puiseux expansions of F above X = 0 in less than O˜(D υ) operations in K, where υ is the valuation of the resultant of F and its partial derivative with respect to Y. To this aim, we use a divide and conquer strategy and replace univariate factorization by dynamic evaluation. As a first main corollary, we compute the irreducible factors of F in K[[X]][Y ] up to an arbitrary precision X N with O˜(D(υ + N)) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with O˜(D 3) arithmetic operations and, if K = Q, with O˜(hD 3) bit operations where h is the logarithmic heigth of F .