22 research outputs found
Radical parametrization of algebraic curves and surfaces
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
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
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
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 -adic cohomology
Galois descent for the gonality of curves
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
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 belongs to the class NP, and we
give an algorithm that computes the stable gonality of a graph in
time.Comment: 15 pages; v2 minor changes; v3 minor change