Carlos Villarino

Covering of surfaces parametrized without projective base points

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.

A FIRST APPROACH TOWARDS NORMAL PARAMETRIZATIONS OF ALGEBRAIC SURFACES

In this paper we analyze the problem of deciding the normality (i.e. the surjectivity) of a rational parametrization of a surface . The problem can be approached by means of elimination theory techniques, providing a proper close subset where surjectivity needs to be analyzed. In general, these direct approaches are unfeasible because is very complicated and its elements computationally hard to manipulate. Motivated by this fact, we study ad hoc computational alternative methods that simplifies . For this goal, we introduce the notion of pseudo-normality, a concept that provides necessary conditions for a parametrization for being normal. Also, we provide an algorithm for deciding the pseudo-normality. Finally, we state necessary and sufficient conditions on a pseudo-normal parametrization to be normal. As a consequence, certain types of parametrizations are shown to be always normal. For instance, pseudo-normal polynomial parametrizations are normal. Moreover, for certain class of parametrizations, we derive an algorithm for deciding the normality.

OPTIMAL REPARAMETRIZATION OF POLYNOMIAL ALGEBRAIC CURVES

In this paper, we present an algorithm for optimally parametrizing polynomial algebraic curves. Let be a polynomial plane algebraic curve given by a polynomial parametrization , where is a finite field extension of a field of characteristic zero. We prove that if is polynomial over , then Weils descente variety associated with is surprisingly simple; it is, in fact, a line. Applying this result we are able to derive an effective algorithm to algebraically optimal reparametrize polynomial algebraic curves.

Reparametrizing swung surfaces over the reals

Let

Missing sets in rational parametrizations of surfaces of revolution

\mathbb {K}\subseteq \mathbb {R}

Original article: Algorithmic detection of hypercircles

K⊆R be a computable subfield of the real numbers (for instance,

Covering rational ruled surfaces

\mathbb {Q}