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
We develop in this article an algorithm that, given a projective curve C, computes a gonal map, that is, a finite morphism from C to P 1 of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality ≤ 4. C ֒→ P g−1 = P(Γ(C, ω C ))
This article gives an algorithm to recover the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry. p
a b s t r a c tWe propose a new lifting and recombination scheme for rational bivariate polynomial factorization that takes advantage of the Newton polytope geometry. We obtain a deterministic algorithm that can be seen as a sparse version of an algorithm of Lecerf, with a polynomial complexity in the volume of the Newton polytope. We adopt a geometrical point of view, the main tool being derived from some algebraic osculation criterion in toric varieties.
We prove a theorem on algebraic osculation and apply our result to the computer algebra problem of polynomial factorization. We consider X a smooth completion of C 2 and D an effective divisor with support the boundary ∂X = X \ C 2 . Our main result gives explicit conditions that are necessary and sufficient for a given Cartier divisor on the subscheme (|D|, O D ) to extend to X. These osculation criteria are expressed with residues. When applied to the toric setting, our result gives rise to a new algorithm for factoring sparse bivariate polynomials which takes into account the geometry of the Newton polytope.
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