13,375 research outputs found

    Computing Puiseux series : a fast divide and conquer algorithm

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    Let F∈K[X,Y]F\in \mathbb{K}[X, Y ] be a polynomial of total degree DD defined over a perfect field K\mathbb{K} of characteristic zero or greater than DD. Assuming FF separable with respect to YY , we provide an algorithm that computes the singular parts of all Puiseux series of FF above X=0X = 0 in less than O~(Dδ)\tilde{\mathcal{O}}(D\delta) operations in K\mathbb{K}, where δ\delta is the valuation of the resultant of FF and its partial derivative with respect to YY. 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 FF in K[[X]][Y]\mathbb{K}[[X]][Y ] up to an arbitrary precision XNX^N with O~(D(δ+N))\tilde{\mathcal{O}}(D(\delta + N )) arithmetic operations. As a second main corollary, we compute the genus of the plane curve defined by FF with O~(D3)\tilde{\mathcal{O}}(D^3) arithmetic operations and, if K=Q\mathbb{K} = \mathbb{Q}, with O~((h+1)D3)\tilde{\mathcal{O}}((h+1)D^3) bit operations using a probabilistic algorithm, where hh is the logarithmic heigth of FF.Comment: 27 pages, 2 figure

    Stability Conditions and Lagrangian Cobordisms

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    In this paper we study the interplay between Lagrangian cobordisms and stability conditions. We show that any stability condition on the derived Fukaya category DFuk(M)D\mathcal{F}uk(M) of a symplectic manifold (M,ω)(M,\omega) induces a stability condition on the derived Fukaya category of Lagrangian cobordisms DFuk(C×M)D\mathcal{F}uk(\mathbb{C} \times M). In addition, using stability conditions, we provide general conditions under which the homomorphism Θ:ΩLag(M)→K0(DFuk(M))\Theta: \Omega_{Lag}(M)\to K_0(D\mathcal{F}uk(M)), introduced by Biran and Cornea, is an isomorphism. This yields a better understanding of how stability conditions affect Θ\Theta and it allows us to elucidate Haug's result, that the Lagrangian cobordism group of T2T^2 is isomorphic to K0(DFuk(T2))K_0(D\mathcal{F}uk(T^2)).Comment: 53 pages, 3 figures, expansions and revisions, improvement of expositio
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