Computing Puiseux series: a fast divide and conquer algorithm

Let F ∈ [ X , Y ] be a polynomial of total degree D defined over a perfect field 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 ˜ ( D δ ) operations in ,...

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Bibliographic Details
Published in:Annales Henri Lebesgue Vol. 4; pp. 1061 - 1102
Main Authors: Poteaux, Adrien, Weimann, Martin
Format: Journal Article
Language:English
Published: UFR de Mathématiques - IRMAR 2021
Subjects:
Commutative Algebra
Computer Science
Mathematics
Symbolic Computation
complexity
Puiseux series
ISSN:2644-9463, 2644-9463
Online Access:Get full text
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Summary:Let F ∈ [ X , Y ] be a polynomial of total degree D defined over a perfect field 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 ˜ ( D δ ) operations in , 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 [ [ 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 = ℚ , with Ø ˜ ( ( h + 1 ) D 3 ) bit operations using probabilistic algorithms, where h is the logarithmic height of F .
ISSN:2644-9463
2644-9463
DOI:10.5802/ahl.97