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 Economía y CompetitividadJunta de ExtremaduraEuropean Regional Development Fun

Computing Birational Polynomial Surface Parametrizations without Base Points

In this paper, we present an algorithm for reparametrizing birational surface parametrizations into birational polynomial surface parametrizations without base points, if they exist. For this purpose, we impose a transversality condition to the base points of the input parametrization

Computing birational polynomial surface parametrizations without base points

In this paper, we present an algorithm for reparametrizing birational surface parametrizations into birational polynomial surface parametrizations without base points, if they exist. For this purpose, we impose a transversality condition to the base points of the input parametrization.Agencia Estatal de Investigació

A Maple package to deal with the birationality of curves and surfaces parametrizations

Maple Conference 2020. Communications in Computer and Information Science book series (CCIS volume 1414)In this paper we present the Maple package Luroth for dealing with the birationality of curves and surfaces parametrizations. The procedures in the package decide whether a given, either curve or surface, parametrization is injective by computing its degree map. In addition, if the parametrization is not injective, it determines a birational reparametrization. For the curve case, the corresponding command always provides an optional answer. For the surface case, not all cases are covered. Nevertheless, we illustrate using Maple some new ideas on how to approach those surface cases not covered in the package.Agencia Estatal de Investigació

Compactifications of rational maps, and the implicit equations of their images

In this paper we give different compactifications for the domain and the codomain of an affine rational map $f$ which parametrizes a hypersurface. We show that the closure of the image of this map (with possibly some other extra hypersurfaces) can be represented by a matrix of linear syzygies. We compactify $\Bbb {A}^{n-1}$ into an $(n-1)$-dimensional projective arithmetically Cohen-Macaulay subscheme of some $\Bbb {P}^N$. One particular interesting compactification of $\Bbb {A}^{n-1}$ is the toric variety associated to the Newton polytope of the polynomials defining $f$. We consider two different compactifications for the codomain of $f$: $\Bbb {P}^n$ and $(\Bbb {P}^1)^n$. In both cases we give sufficient conditions, in terms of the nature of the base locus of the map, for getting a matrix representation of its closed image, without involving extra hypersurfaces. This constitutes a direct generalization of the corresponding results established in [BuseJouanolou03], [BuseChardinJouanolou06], [BuseDohm07], [BotbolDickensteinDohm09] and [Botbol09].Comment: 2 images, 28 pages. To appear in Journal of Pure and Applied Algebr

Sufficient conditions for the surjectivity of radical curve parametrizations

J. Caravantes 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 Fondo Europeo de Desarrollo Regional (GR21055).En este artículo se introduce la noción de parametrización radical suprayectiva y se dan condiciones suficientes para que una parametrización radical de una curva sea suprayectiva.In this paper, we introduce the notion of surjective radical parametrization and we prove sufficient conditions for a radical curve parametrization to be surjective.Agencia Estatal de Investigació