2,733 research outputs found

    Computing canonical heights using arithmetic intersection theory

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    For several applications in the arithmetic of abelian varieties it is important to compute canonical heights. Following Faltings and Hriljac, we show how the canonical height on the Jacobian of a smooth projective curve can be computed using arithmetic intersection theory on a regular model of the curve in practice. In the case of hyperelliptic curves we present a complete algorithm that has been implemented in Magma. Several examples are computed and the behavior of the running time is discussed.Comment: 29 pages. Fixed typos and minor errors, restructured some sections. Added new Example

    Toric Intersection Theory for Affine Root Counting

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    Given any polynomial system with fixed monomial term structure, we give explicit formulae for the generic number of roots with specified coordinate vanishing restrictions. For the case of affine space minus an arbitrary union of coordinate hyperplanes, these formulae are also the tightest possible upper bounds on the number of isolated roots. We also characterize, in terms of sparse resultants, precisely when these upper bounds are attained. Finally, we reformulate and extend some of the prior combinatorial results of the author on which subsets of coefficients must be chosen generically for our formulae to be exact. Our underlying framework provides a new toric variety setting for computational intersection theory in affine space minus an arbitrary union of coordinate hyperplanes. We thus show that, at least for root counting, it is better to work in a naturally associated toric compactification instead of always resorting to products of projective spaces

    Partial Degree Formulae for Plane Offset Curves

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    In this paper we present several formulae for computing the partial degrees of the defining polynomial of the offset curve to an irreducible affine plane curve given implicitly, and we see how these formulae particularize to the case of rational curves. In addition, we present a formula for computing the degree w.r.t the distance variable.Comment: 24 pages, no figure

    Curves having one place at infinity and linear systems on rational surfaces

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    Denoting by Ld(m0,m1,...,mr){\mathcal L}_d(m_0,m_1,...,m_r) the linear system of plane curves passing through r+1r+1 generic points p0,p1,...,prp_0,p_1,...,p_r of the projective plane with multiplicity mim_i (or larger) at each pip_i, we prove the Harbourne-Hirschowitz Conjecture for linear systems Ld(m0,m1,...,mr){\mathcal L}_d(m_0,m_1,...,m_r) determined by a wide family of systems of multiplicities m=(mi)i=0r\bold{m}=(m_i)_{i=0}^r and arbitrary degree dd. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system m\bold{m} and we give its exact value when m\bold{m} is in the above family. To do that, we prove an H1H^1-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.Comment: This is a revised version of a preprint of 200

    Computing functions on Jacobians and their quotients

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    We show how to efficiently compute functions on jacobian varieties and their quotients. We deduce a quasi-optimal algorithm to compute (l,l)(l,l) isogenies between jacobians of genus two curves
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