Carlos Andradas

Plotting missing points and branches of real parametric curves

This paper is devoted to the study (from the theoretic and algorithmic point of view) of the existence of points and branches non-reachable by a parametric representation of a rational algebraic curve (in n-dimensional space) either over the field of complex numbers or over the field of real numbers. In particular, we generalize some of the results on missing points in (J. Symbolic Comput. 33, 863–885, 2002) to the case of space curves. Moreover, we introduce for the first time and we solve the case of missing branches. Another novelty is the emphasis on topological conditions over the curve for the existence of missing points and branches. Finally, we would like to point out that, by developing an “ad hoc” and simplified theory of valuations for the case of parametric curves, we approach in a new and unified way the analysis of the missing points and branches, and the proposal of the algorithmic solution to these problems.

Base field restriction techniques for parametric curves

Given a variety V, implicitly defined over an algebraic separable field extension k(alpha), A. Weil [5] developed a restriction technique (called by him a descente method),that associates to V a suitable k-variety W, such that many properties of V can be analyzed by merely looking at W, that is, by descending to the base field k. In this paper we present a parametric counterpart, for curves, of Weils construction. As an application, we state some simple algorithmic criteria over the variety W that translate, for instance, the k-definability of a parametric curve V, or the existence of an infinite number of L-rational points in V.

A relatively optimal rational space curve reparametrization algorithm through canonical divisors

Let K be a given computable field of characteristic zero and let ILbe a finite field extension of K, with algebraic closure F. Assume a rational parametrization P(t) E L(t)nof some irreducible curve C in the affine n-space over F is also given. In this paper we will show, first, how to decide -without imp]icitization algorithms– whether the given curve C is definable (by a set of equations with coefficients) over K; and, if this is the case, we will determine –without computing the implicit, equation set and then using parametrization techniques a reparametrizatiou of P(t) over the smallest possible field extension of K; that is, over a field extension of K of degree at most two.

On the simplification of the coefficients of a parametrization

This paper deals with the problem of finding, for a given parametrization of an algebraic variety V of arbitrary dimension, another parametrization with coefficients over a smaller field. We proceed adapting, to the parametric case, a construction introduced by A. Weil for implicitly given varieties. We find that this process leads to the consideration of new varieties of a particular kind (ultraquadrics, in the terminology of this paper) in order to check, algorithmically, several interesting properties of the given variety V, such as the property of being reparametrizable over the smaller field.

Separation of semialgebraic sets

We study the problem of deciding whether two disjoint semialgebraic sets of an algebraic variety over R are separable by a polynomial. For that we isolate a dense subfamily of spaces of orderings, named geometric, which suffice to test separation and that reduce the problem to the study of the behaviour of the semialgebraic sets in their boundary. Then we derive several characterizations for the generic separation, among which there is a geometric criterion that can be tested algorithmically. Finally we show how to check recursively whether we can pass from generic separation to separation, obtaining a decision procedure for solving the problem.

Reparametrizing swung surfaces over the reals

Let

STABILITY INDEX OF CLOSED SEMIANALYTIC SET GERMS

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

Specialization chains of real valuation rings

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