2,653 research outputs found
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
Computing Hypercircles by Moving Hyperplanes
Let K be a field of characteristic zero, alpha algebraic of degree n over K.
Given a proper parametrization psi of a rational curve C, we present a new
algorithm to compute the hypercircle associated to the parametrization psi. As
a consequence, we can decide if the curve C is defined over K and, if not, to
compute the minimum field of definition of C containing K. The algorithm
exploits the conjugate curves of C but avoids computation in the normal closure
of K(alpha) over K.Comment: 16 page
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
A combinatorial Li-Yau inequality and rational points on curves
We present a method to control gonality of nonarchimedean curves based on graph theory. Let k denote a complete nonarchimedean valued field.We first prove a lower bound for the gonality of a curve over the algebraic closure of k in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some âvolumeâ of the original graph; this can be seen as a substitute for graphs of the LiâYau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally,we apply the results to give a lower bound for the gonality of arbitraryDrinfeld modular curves over finite fields and for general congruence subgroups Î of Î (1) that is linear in the index [Î (1) : Î ], with a constant that only depends on the residue field degree and the degree of the chosen âinfiniteâ place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian
Covering rational surfaces with rational parametrization images
Let S be a rational projective surface given by means of a projective rational parametrization
whose base locus satisfies a mild assumption. In this paper we present an algorithm that provides
three rational maps f , g, h : A2 S â P
n
such that the union of the three images covers S. As a
consequence, we present a second algorithm that generates two rational maps f , gË : A2 S, such
that the union of its images covers the affine surface S â© An
. In the affine case, the number of rational
maps involved in the cover is in general optimal.Ministerio de Ciencia, InnovaciĂłn y UniversidadesJ. Caravantes, J.R. Sendra and C. Villarino belong to the Research Group ASYNACS (Ref. CT-CE2019/683). D. Sevilla is a member of the research group GADAC and is partially supported by Junta de Extremadura and FEDER funds (group FQM024)
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