3,377 research outputs found
Formal Desingularization of Surfaces - The Jung Method Revisited -
In this paper we propose the concept of formal desingularizations as a
substitute for the resolution of algebraic varieties. Though a usual resolution
of algebraic varieties provides more information on the structure of
singularities there is evidence that the weaker concept is enough for many
computational purposes. We give a detailed study of the Jung method and show
how it facilitates an efficient computation of formal desingularizations for
projective surfaces over a field of characteristic zero, not necessarily
algebraically closed. The paper includes a generalization of Duval's Theorem on
rational Puiseux parametrizations to the multivariate case and a detailed
description of a system for multivariate algebraic power series computations.Comment: 33 pages, 2 figure
Involutions of polynomially parametrized surfaces
We provide an algorithm for detecting the involutions leaving a surface
defined by a polynomial parametrization invariant. As a consequence, the
symmetry axes, symmetry planes and symmetry center of the surface, if any, can
be determined directly from the parametrization, without computing or making
use of the implicit representation. The algorithm is based on the fact, proven
in the paper, that any involution of the surface comes from an involution of
the parameter space (the real plane, in our case); therefore, by determining
the latter, the former can be found. The algorithm has been implemented in the
computer algebra system Maple 17. Evidence of its efficiency for moderate
degrees, examples and a complexity analysis are also given
Implicitization of Bihomogeneous Parametrizations of Algebraic Surfaces via Linear Syzygies
We show that the implicit equation of a surface in 3-dimensional projective
space parametrized by bi-homogeneous polynomials of bi-degree (d,d), for a
given positive integer d, can be represented and computed from the linear
syzygies of its parametrization if the base points are isolated and form
locally a complete intersection
Symmetry Detection of Rational Space Curves from their Curvature and Torsion
We present a novel, deterministic, and efficient method to detect whether a
given rational space curve is symmetric. By using well-known differential
invariants of space curves, namely the curvature and torsion, the method is
significantly faster, simpler, and more general than an earlier method
addressing a similar problem. To support this claim, we present an analysis of
the arithmetic complexity of the algorithm and timings from an implementation
in Sage.Comment: 25 page
Nash Problem for quotient surface singularities
We give an affirmative answer to Nash Problem for quotient surface
singularities, in particular for the icosahedral singularity .Comment: 25 pages with 5 figures. This is part of the author\'s PhD thesi
Generic uniqueness of least area planes in hyperbolic space
We study the number of solutions of the asymptotic Plateau problem in H^3. By
using the analytical results in our previous paper, and some topological
arguments, we show that there exists an open dense subset of C^3 Jordan curves
in S^2_{infty}(H^3) such that any curve in this set bounds a unique least area
plane in H^3.Comment: This is the version published by Geometry & Topology on 27 April 2006
(V3: typesetting corrections
Topology of 2D and 3D Rational Curves
In this paper we present algorithms for computing the topology of planar and
space rational curves defined by a parametrization. The algorithms given here
work directly with the parametrization of the curve, and do not require to
compute or use the implicit equation of the curve (in the case of planar
curves) or of any projection (in the case of space curves). Moreover, these
algorithms have been implemented in Maple; the examples considered and the
timings obtained show good performance skills.Comment: 26 pages, 19 figure
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