International audienc

Let F ∈ K[X, Y ] be a polynomial of total degree D defined over a perfect field K of characteristic zero or greater than D. Assuming F separable with respect to Y , we provide an algorithm that computes all singular parts of Puiseux series of F above X = 0 in an expected O (D δ) operations in K, where δ 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 factorisation 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(δ + 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 ((h + 1)D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F .Résumé. -Soit F ∈ K[X, Y ] un polynôme de degré total D défini au dessus d'un corps parfait K de caractéristique zéro ou plus grande que D. Sous l'hypothèse que F est séparable par rapport à la variable Y , nous décrivons un algorithme qui calcule l'ensemble des parties singulières des séries de Puiseux de F au-dessus de X = 0 avec un nombre moyen d'opérations