Good reduction of Puiseux series and applications

Poteaux, Adrien; Rybowicz, Marc

doi:10.1016/j.jsc.2011.08.008

Cited by 17 publications

(18 citation statements)

References 31 publications

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“…As computing Puiseux expansions can be costly, we use a modular-numeric algorithm. It was first described in [Pot07] and improved in [Pot08] (the modular part of the algorithm is also described in [PR08,PRb,PRa]). All details of our monodromy algorithm can be found in [Pot07] and [Pot08].…”

Section: Connection Methodmentioning

confidence: 99%

Computing monodromy via continuation methods on random Riemann surfaces

Galligo

Poteaux²

2011

Theoretical Computer Science

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We consider a Riemann surface X defined by a polynomial f (x, y) of degree d, whose coefficients are chosen randomly. Hence, we can suppose that X is smooth, that the discriminant δ(x) of f has d(d − 1) simple roots, ∆, and that δ(0) = 0 i.e. the corresponding fiber has d distinct points {y1,. .. , y d }. When we lift a loop 0 ∈ γ ⊂ C − ∆ by a continuation method, we get d paths in X connecting {y1,. .. , y d }, hence defining a permutation of that set. This is called monodromy. Here we present experimentations in Maple to get statistics on the distribution of transpositions corresponding to loops around each point of ∆. Multiplying families of "neighbor" transpositions, we construct permutations and the subgroups of the symmetric group they generate. This allows us to establish and study experimentally two conjectures on the distribution of these transpositions and on transitivity of the generated subgroups. Assuming that these two conjectures are true, we develop tools allowing fast probabilistic algorithms for absolute multivariate polynomial factorization, under the hypothesis that the factors behave like random polynomials whose coefficients follow uniform distributions.

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Section: Connection Methodmentioning

confidence: 99%

Computing monodromy via continuation methods on random Riemann surfaces

Galligo

Poteaux²

2011

Theoretical Computer Science

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“…Our interest in the finite field case stemmed from a modular reduction method that we proposed to avoid coefficient swell in the number field case [39,41]. In this context, the condition p > dY may always be enforced and is part of our good reduction criterion.…”

Section: Introductionmentioning

confidence: 99%

“…Merle and Henry [27], then Teitelbaum [46] studied the arithmetic complexity of the resolution of the singularity at the origin defined by F (X, Y ) = 0, a process tightly related to Puiseux series [9]. We have commented on these works and explained why we prefer to stick to the Newton-Puiseux algorithm in [40,41].…”

Section: Introductionmentioning

confidence: 99%

Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

Poteaux

Rybowicz

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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International audienceLet L be a field of characteristic p with q elements and F ∈ L[X, Y ] be a polynomial with p > deg Y (F) and total degree d. In [40], we showed that rational Puiseux series of F above X = 0 could be computed with an expected number of O˜d 3 log q) arithmetic operations in L. In this paper, we reduce this bound to O˜og q) using Hensel lifting and changes of variables in the Newton-Puiseux algorithm that give a better control of the number of steps. The only asymptotically fast algorithm required is polynomial multiplication over finite fields. This approach also allows to test the irreducibility of F in L[[X]][Y ] with Oõperations in L. Finally, we describe a method based on structured bivariate multiplication [34] that may speed up computations for some input

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“…For the leading terms of the Puiseux series, we rely on tropical methods [9], and in particular on the constructive proof of the fundamental theorem of tropical algebraic geometry [21], see also [23] and [28]. Computer algebra methods for Puiseux series in two dimensions can be found in [29]. Our contributions.…”

Section: Preliminariesmentioning

confidence: 99%

The method of Gauss–Newton to compute power series solutions of polynomial homotopies

Bliss

Verschelde

2018

Linear Algebra and its Applications

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We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.

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Good reduction of Puiseux series and applications

Cited by 17 publications

References 31 publications

Computing monodromy via continuation methods on random Riemann surfaces

Computing monodromy via continuation methods on random Riemann surfaces

Improving Complexity Bounds for the Computation of Puiseux Series over Finite Fields

The method of Gauss–Newton to compute power series solutions of polynomial homotopies

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