22 research outputs found

    Radical parametrization of algebraic curves and surfaces

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    The first author is a member of the Research Group asynacs (Ref. ccee2011/r34).Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; offsets and conchoids; etc.) There are many results on rational parametrization, in particular in the curve case, but the class of such objects is relatively small. If we allow root extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves can be parametrized with one square root). We will describe the basics and the state of the art of the problem of parametrization of curves and surfaces by radicals.Ministerio de EconomĂ­a y CompetitividadAustrian Science Fund (FWF

    Radical parametrization of algebraic curves and surfaces

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    Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; offsets and conchoids; etc.) There are many results on rational parametrization, in particular in the curve case, but the class of such objects is relatively small. If we allow root extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves can be parametrized with one square root). We will describe the basics and the state of the art of the problem of parametrization of curves and surfaces by radicals.Junta de Extremadura and FEDER fundsThis contribution is partially supported by the Ministerio de Econom´ıa y Competitividad under the project MTM2011-25816-C02-01, by the Austrian Science Fund (FWF) P22766-N18, and by Junta de Extremadura and FEDER funds. The first author is a member of the of the Research Group asynacs (Ref. ccee2011/r34)

    Algebraic and algorithmic aspects of radical parametrizations

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    In this article algebraic constructions are introduced in order to study the variety defined by a radical parametrization (a tuple of functions involving complex numbers, n variables, the four field operations and radical extractions). We provide algorithms to implicitize radical parametrizations and to check whether a radical parametrization can be reparametrized into a rational parametrization.Ministerio de EconomĂ­a y CompetitividadJunta de ExtremaduraEuropean Regional Development Fun

    Point counting on curves using a gonality preserving lift

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    We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using pp-adic cohomology

    Galois descent for the gonality of curves

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    We determine conditions for the invariance of the gonality under base extension, depending on the numeric invariants of the curve. More generally, we study the Galois descent of morphisms of curves to Brauer-Severi varieties, and also of rational normal scrolls

    Stable gonality is computable

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    Stable gonality is a multigraph parameter that measures the complexity of a graph. It is defined using maps to trees. Those maps, in some sense, divide the edges equally over the edges of the tree; stable gonality asks for the map with the minimum number of edges mapped to each edge of the tree. This parameter is related to treewidth, but unlike treewidth, it distinguishes multigraphs from their underlying simple graphs. Stable gonality is relevant for problems in number theory. In this paper, we show that deciding whether the stable gonality of a given graph is at most a given integer kk belongs to the class NP, and we give an algorithm that computes the stable gonality of a graph in O((1.33n)nmmpoly(n,m))O((1.33n)^nm^m \text{poly}(n,m)) time.Comment: 15 pages; v2 minor changes; v3 minor change
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