Carlos Javier Peláez Villarino - Academia.edu (original) (raw)

Papers by Carlos Javier Peláez Villarino

Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas

We prove that any smooth rational projective surface over the field of complex numbers has an ope... more We prove that any smooth rational projective surface over the field of complex numbers has an open covering consisting of 3 subsets isomorphic to affine planes.

arXiv (Cornell University), Mar 13, 2020

We present an algorithm that transforms, if possible, a given ODE or PDE with radical function co... more We present an algorithm that transforms, if possible, a given ODE or PDE with radical function coefficients into one with rational coefficients by means of a rational change of variables. It also applies to systems of linear ODEs. It is based on previous work on reparametrization of radical algebraic varieties.

Mathematics, 2021

Let S be a rational projective surface given by means of a projective rational parametrization wh... more 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⊂Pn such that the union of the three images covers S. As a consequence, we present a second algorithm that generates two rational maps f,g˜:A2⇢S, such that the union of its images covers the affine surface S∩An. In the affine case, the number of rational maps involved in the cover is in general optimal.

Journal of Algebra, 2018

In this paper we show that not all affine rational complex surfaces can be parametrized birationa... more In this paper we show that not all affine rational complex surfaces can be parametrized birational and surjectively. For this purpose, we prove that, if S is an affine complex surface whose projective closure is smooth, a necessary condition for S to admit a birational surjective parametrization from an open subset of the affine complex plane is that the infinity curve of S must contain at least one rational component. As a consequence of this result we provide examples of affine rational surfaces that do not admit birational surjective parametrizations.

Computer Aided Geometric Design, 2017

In this article algebraic constructions are introduced in order to study the variety defined by a... more 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.

Mathematics of Computation, 2017

We present an algorithm that covers any given rational ruled surface with two rational parametriz... more We present an algorithm that covers any given rational ruled surface with two rational parametrizations. In addition, we present an algorithm that transforms any rational surface parametrization into a new rational surface parametrization without affine base points and such that the degree of the corresponding maps is preserved.

Computer-Aided Design, 2015

Parametric representations do not cover, in general, the whole geometric object that they paramet... more Parametric representations do not cover, in general, the whole geometric object that they parametrize. This can be a problem in practical applications. In this paper we analyze the question for surfaces of revolution generated by real rational profile curves, and we describe a simple small superset of the real zone of the surface not covered by the parametrization. This superset consists, in the worst case, of the union of a circle and the mirror curve of the profile curve.

Given a rational projective parametrization of a rational surface, we decompose (with the excepti... more Given a rational projective parametrization of a rational surface, we decompose (with the exception of the parametrization base points) the projective plane of parameters as union of sets such that in each of these sets the multiplicity of the achieved points on the surface is invariant.

Given the classical rational parametrization of a surface of revolution, generated by rotating a ... more Given the classical rational parametrization of a surface of revolution, generated by rotating a rational curve around the z-axis, we determine a superset containing all the possible points of the surface non-reachable by the parametrization; that is a critical set of the parametrization.

Journal of Symbolic Computation, 2016

We determine necessary and sufficient conditions for a tubular surface to be swung, and viceversa... more We determine necessary and sufficient conditions for a tubular surface to be swung, and viceversa. From these characterizations, we derive two symbolic algorithms. The first one decides whether a given implicit equation, of a tubular surface, admits a swung parametrization and, in the affirmative case, it outputs such a parametrization. The second one decides whether a given swung surface parametrization is a tubular surface and, in the affirmative case, it outputs the implicit equation.

Applicable Algebra in Engineering, Communication and Computing, 2014

The concept of ultraquadric has been introduced by the authors as a tool to algorithmically solve... more The concept of ultraquadric has been introduced by the authors as a tool to algorithmically solve the problem of simplifying the coefficients of a given rational parametrization in K(α)(t1,. .. , tn) of an algebraic variety of arbitrary dimension over a field extension K(α). In this context, previous work in the one-dimensional case has shown the importance of mastering the geometry of 1-dimensional ultraquadrics (hypercircles). In this paper we study, for the first time, the properties of some higher dimensional ultraquadrics, namely, those associated to automorphisms in the field K(α)(t1,. .. , tn), defined by linear rational (with common denominator) or by polynomial (with inverse also polynomial) coordinates. We conclude, among many other observations, that ultraquadrics related to polynomial automorphisms can be characterized as varieties K−isomorphic to linear varieties, while ultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a blowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric variety. We conclude with some further details about the real-complex, 2dimensional case, showing, for instance, that this family of ultraquadrics can be presented as a collection of ruled surfaces described by pairs of hypercircles.

Proceedings of the 39th International Symposium on Symbolic and Algebraic Computation - ISSAC '14, 2014

We prove that every affine rational surface, parametrized by means of an affine rational parametr... more We prove that every affine rational surface, parametrized by means of an affine rational parametrization without projective base points, can be covered by at most three parametrizations. Moreover, we give explicit formulas for computing the coverings. We provide two different approaches: either covering the surface with a surface parametrization plus a curve parametrization plus a point, or with the original parametrization plus two surface reparametrizations of it.

Lecture Notes in Computer Science, 2015

This paper deals with the decision problem of the surjectivity of a rational surface parametrizat... more This paper deals with the decision problem of the surjectivity of a rational surface parametrization. We give sufficient conditions for a parametrization to be surjective, and we describe different families of parametrizations that satisfy these criteria. In addition, we consider the problem of computing a superset of the points not covered by the parametrization. In this context, we report on the case of parametrizations without projective base points and we analyze the particular case of rational ruled surfaces.

Mathematics of Computation, 2014

Given a rational projective parametrization P(s, t, v) of a rational projective surface S we pres... more Given a rational projective parametrization P(s, t, v) of a rational projective surface S we present an algorithm such that, with the exception of a finite set (maybe empty) B of projective base points of P, decomposes the projective parameter plane as P 2 (K) \ B = k=1 S k such that, if (s 0 : t 0 : v 0) ∈ S k , then P(s 0 , t 0 , v 0) is a point of S of multiplicity k.

Proceedings of the 2004 international symposium on Symbolic and algebraic computation, 2004

Abstract This paper deals with a remarkable class of curves (in general r-space) that the two fir... more Abstract This paper deals with a remarkable class of curves (in general r-space) that the two first authors have named&amp;amp;quot; hypercircles&amp;amp;quot;(see [2]). As shown there, such curves appear in the CAD context, when aiming towards finding a parametric representation with simpler ...

Actas de AGGM, 2006

Let K be a characteristic zero field, α algebraic of degree n over K and F the algebraic closure ... more Let K be a characteristic zero field, α algebraic of degree n over K and F the algebraic closure of K. Let C be a rational curve defined over K given by a proper parametrization ... C ≃ (η1(t),...,ηm(t)), ηi (t) ∈ K(α)(t) ... Is there a parametrization of C over K? It is the case if ...

Journal of Algebra and its Applications, 2002

Abstract: In this paper we deal with the problem of computing algebraically optimal parametrizati... more Abstract: In this paper we deal with the problem of computing algebraically optimal parametrizations for quasi-polynomial algebraic curves. The algebraic optimality parametrization problem consists in computing rational parametrizations which ...

Mathematics of Computation, 2009

This paper deals with a family of spatial rational curves that were introduced in [4], under the ... more This paper deals with a family of spatial rational curves that were introduced in [4], under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line ("the K-axis") in a n-degree finite algebraic extension K(α) ≈ K n under the transformation at+b ct+d : K(α) → K(α). The aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via K-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many K-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section. * The authors are partially supported by the project MTM2005-08690-CO2-01/02 "Ministerio de Educación y Ciencia".

Mathematics and Computers in Simulation, 2011

In the algebraically optimal reparametrization problem, one of the possible approaches deals with... more In the algebraically optimal reparametrization problem, one of the possible approaches deals with computing a parametric variety of Weyl and checking whether this variety is a hypercircle. Here, algorithms to detect whether a curve given parametrically is a ...