Computing Puiseux series: a fast divide and conquer algorithm

Abstract

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

Using approximate roots for irreducibility and equi-singularity issues in K[[x]][y]

“…If F is irreducible, we have moreover v x (F y (S)) = δ d for any Puiseux series S of F . Then, [25,Lemma 6] and [25, Corollary 4] prove that computations can be made modulo x 2δ d +1 , leading to an expected number of operations over K bounded by O((δ+1) d) [25,Proposition 18]. Moreover, as mentionned in the conclusion of [25], this complexity estimates is sharp.…”

“…Factorization in K[[x]][y] (and a fortiori irreducibility test) is an important issue in the algorithmic of algebraic curves, both for local aspects (studying plane curves singularities) and for global aspects (e.g. computing integral basis of function fields [31], computing the geometric genus [25], factoring polynomials in K[x, y] taking advantage of critical fibers [34], etc). Probably the most classical approach for factoring polynomials in K[[x]][y] is derived from the Newton-Puiseux algorithm, as a combination of blow-ups (monomial transforms and shifts) and Hensel liftings.…”

“…[6, 7, 21-25, 27, 33] and the references therein). Up to our knowledge, the best current arithmetic complexity was obtained in [25], using a divide and conquer strategy leading to a fast Newton-Puiseux algorithm (hence an irreducibility test) which computes the singular parts of all Puiseux series above x = 0 in an expected O˜(d δ) operations over K. There exists also other methods for factorization, as the Montes algorithm which allow to factor polynomials over general local fields [14,19] with no assumptions on the characteristic of the residue field. Similarly to the algorithms we present in this paper, Montes et al compute higher order Newton polygons and boundary polynomials from the Φ-adic expansion of F , where Φ is a sequence of some well-chosen polynomials which is updated at each step of the algorithm.…”

“…At the first call, we assume F monic and we rather consider G = F in (2). We call H the Abhyankar transform of F , and denote Abhyankar(F ) the subroutine computing it (with infinite precision in what follows, but with a suitable finite precision in practice, see [25,Section 3.3]). The polynomial H has now no terms of degree N − 1, ensuring qℓ > 1 at line 4.…”

“…Remark 1. The (q, m)-sequence of ARNP is not the same as the (q, m)-sequence of RNP, but can be deduced from it [25,Remark 5 and Example 2]. It contains enough information for computing the characteristic exponents of F ; see Section 8.…”