2,633 research outputs found
Regularity criteria for the topology of algebraic curves and surfaces
In this paper, we consider the problem of analysing the shape of an object defined by polynomial equations in a domain. We describe regularity criteria which allow us to determine the topology of the implicit object in a box from information on the boundary of this box. Such criteria are given for planar and space algebraic curves and for algebraic surfaces. These tests are used in subdivision methods in order to produce a polygonal approximation of the algebraic curves or surfaces, even if it contains singular points. We exploit the representation of polynomials in Bernstein basis to check these criteria and to compute the intersection of edges or facets of the box with these curves or surfaces. Our treatment of singularities exploits results from singularity theory such as an explicit Whitney stratification or the local conic structure around singularities. A few examples illustrate the behavior of the algorithms
Analytic curves in algebraic varieties over number fields
We establish algebraicity criteria for formal germs of curves in algebraic
varieties over number fields and apply them to derive a rationality criterion
for formal germs of functions, which extends the classical rationality theorems
of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to
arbitrary algebraic curves over a number field.
The formulation and the proof of these criteria involve some basic notions in
Arakelov geometry, combined with complex and rigid analytic geometry (notably,
potential theory over complex and -adic curves). We also discuss geometric
analogues, pertaining to the algebraic geometry of projective surfaces, of
these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor
of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200
Shrinkwrapping and the taming of hyperbolic 3-manifolds
We introduce a new technique for finding CAT(-1) surfaces in hyperbolic
3-manifolds. We use this to show that a complete hyperbolic 3-manifold with
finitely generated fundamental group is geometrically and topologically tame.Comment: 60 pages, 7 figures; V3: incorporates referee's suggestions,
references update
Implicitization using approximation complexes
We present a method for the implicitization problem that goes back to the
work of Sederberg and Chen. The formalism we use, with approximation complexes
as a key ingredient, is due to Jean-Pierre Jouanolou and was explained in
details in this context in his joint work with Laurent Buse. Most of this note
is dedicated to presenting the method, the geometric ideas behind it and the
tools from commutative algebra that are needed. In the last section, we give
the most advanced results we know related to this approach.Comment: 13 page
An embedding technique for the solution of reaction-fiffusion equations on algebraic surfaces with isolated singularities
In this paper we construct a parametrization-free embedding technique for numerically evolving reaction-diffusion PDEs defined on algebraic curves that possess an isolated singularity. In our approach, we first desingularize the curve by appealing to techniques from algebraic geometry.\ud
We create a family of smooth curves in higher dimensional space that correspond to the original curve by projection. Following this, we pose the analogous reaction-diffusion PDE on each member of this family and show that the solutions (their projection onto the original domain) approximate the solution of the original problem. Finally, we compute these approximants numerically by applying the Closest Point Method which is an embedding technique for solving PDEs on smooth surfaces of arbitrary dimension or codimension, and is thus suitable for our situation. In addition, we discuss the potential to generalize the techniques presented for higher-dimensional surfaces with multiple singularities
Isotopic Equivalence from Bezier Curve Subdivision
We prove that the control polygon of a Bezier curve B becomes homeomorphic
and ambient isotopic to B via subdivision, and we provide closed-form formulas
to compute the number of iterations to ensure these topological
characteristics. We first show that the exterior angles of control polygons
converge exponentially to zero under subdivision.Comment: arXiv admin note: substantial text overlap with arXiv:1211.035
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