Computing Puiseux series: a fast divide and conquer algorithm
Annales Henri Lebesgue, Volume 4 (2021), pp. 1061-1102.

Let F𝕂[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 Ø ˜(Dδ ) 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 Ø ˜(D(δ +N)) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by F with Ø ˜(D 3 ) arithmetic operations and, if 𝕂=, with Ø ˜((h+1)D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F.

Soit F𝕂[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 sur 𝕂 borné par Ø ˜(Dδ ), où δ est la valuation du résultant de F et sa dérivée partielle en Y. Pour se faire, nous utilisons une stratégie de type “diviser pour régner” et nous remplaçons l’utilisation de factorisation univariée par l’évaluation dynamique. Comme premier corollaire principal, nous calculons les facteurs irréductibles de F dans 𝕂[[X]][Y] à precision X N en Ø ˜(D(δ +N)) opérations arithmétiques. Comme second corollaire, nous calculons le genre de la courbe algébrique plane définie par F en Ø ˜(D 3 ) opérations arithmétiques, et, si 𝕂=, en Ø ˜((h+1)D 3 ) opérations binaires via des algorithmes probabilites, où h est la taille logarithmique de F.

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Accepted:
Published online:
DOI: 10.5802/ahl.97
Classification: 14B05, 14H20, 14Q20, 68Q25
Mots-clés : Puiseux series, complexity, dynamic evaluation
Poteaux, Adrien 1; Weimann, Martin 2

1 CRIStAL, Université de Lille, UMR CNRS 9189, Bâtiment Esprit, 59655 Villeneuve d’Ascq, (France)
2 GAATI: Current delegation, Université de Polynésie Française, UMR CNRS 6139, BP 6570, 98702 Faa’a, Polynésie Française, (France) Permanent position: LMNO, University of Caen-Normandie, BP 5186, 14032 Caen Cedex, (France)
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Poteaux, Adrien; Weimann, Martin. Computing Puiseux series: a fast divide and conquer algorithm. Annales Henri Lebesgue, Volume 4 (2021), pp. 1061-1102. doi : 10.5802/ahl.97. http://www.numdam.org/articles/10.5802/ahl.97/

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