Preprints, Working Papers, ... Year : 2017

A dichotomic Newton-Puiseux algorithm using dynamic evaluation

Abstract

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 .

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Dates and versions

hal-01578214 , version 1 (28-08-2017)
hal-01578214 , version 2 (03-01-2018)
hal-01578214 , version 3 (03-12-2018)

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  • HAL Id : hal-01578214 , version 2

Cite

Adrien Poteaux, Martin Weimann. A dichotomic Newton-Puiseux algorithm using dynamic evaluation. 2017. ⟨hal-01578214v2⟩
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