Computing Puiseux series: a fast divide and conquer algorithm
Résumé
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 Puiseux series of F above X = 0 in less than $O˜(D δ)$ 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 .
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