In this paper we study systems of autonomous algebraic ODEs in several differential indeterminates. We develop a notion of algebraic dimension of such systems by considering them as algebraic systems. Afterwards we apply differential elimination and analyze the behavior of the dimension in the resulting Thomas decomposition. For such systems of algebraic dimension one, we show that all formal Puiseux series solutions can be approximated up to an arbitrary order by convergent solutions. We show that the existence of Puiseux series and algebraic solutions can be decided algorithmically. Moreover, we present a symbolic algorithm to compute all algebraic solutions. The output can either be represented by triangular systems or by their minimal polynomials.

We present a deterministic algorithm for deciding if a polynomial ideal, with coefficients in an algebraically closed field K of characteristic zero, of which we know just some very limited data, namely: the number n of variables, and some upper bound for the geometric degree of its zero set in Kn, is or not the zero ideal. The algorithm performs just a finite number of decisions to check wheather a point is or not in the zero set of the ideal. Moreover, we extend this technique to test, in the same fashion, if the elimination of some variables in the given ideal yields or not the zero ideal. Finally, the role of this technique in the context of automated theorem proving of elementary geometry statements, is presented, with references to recent documents describing the excellent performance of the already existing prototype version, implemented in GeoGebra.

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.

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present an algorithm for parametrizing by radicals any irreducible plane curve of degree d having at least a point of multiplicity d−r, with 1≤r≤4 and, as a consequence, every irreducible plane curve of degree d≤5 and every irreducible singular plane curve of degree 6.

In this paper, we present formulae for the computation of the partial degrees w.r.t. each variable of the implicit equation of a rational surface given by means of a proper parametrization. Moreover, when the parametrization is not proper we give upper bounds. These formulae generalize the results in [17] to the surface case, and they are based on the computation of the degree of the rational maps induced by the projections, onto the coordinate planes of the three dimensional space, of the input surface parametrization. In addition, using the results presented in [9] and [10], the formulae simply involve the computation of the degree of univariate polynomials directed determined from the parametrization by means of some univariate resultants and some polynomial gcds.

This paper deals with a remarkable class of curves (in general r-space) that the two first authors have named "hypercircles" (see [2]). As shown there, such curves appear in the CAD context, when aiming towards finding a parametric representation with simpler coefficients (i.e. without algebraic numbers) for a given parametric curve. In fact, it turns out that the crucial point to solve the simplification problem in general is to solve this same problem for hypercircles [2]. Here we present an algorithm that, for a given parametrization of a hypercircle U, over an algebraic extension, namely, φ(t) ∈ K(α)(t)r, computes the linear fraction over K(α)(t) that generates this hypercircle (and, in particular, a parametrization of the curve over K).

The degree of a rational map measures how often the map covers the image variety. In particular, when the rational map is a parametrization, the degree measures how often the parametrization traces the image. We show how the degree of rational maps between algebraic curves can be determined efficiently. In the process, we also give a complete proof of Sederberg's approach for making a parametrization proper.

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 Weil's 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.

In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational,

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.

In this paper, we investigate the problem of the rank and the determinant of Hankel matrices over Z[zl,..., z,].A modular algorithm for determining the rank of a Hankel matrix with entries that are multiwwiate poly-

The problem of computing the number of distinct real roots of a real polynomial can be solved analyzing the sign variations of the sequence of principal minors of the Hankel matrix associated with the given polynomial. In this paper, we present a modular algorithm to achieve this goal. In this approach, the principal minors sequence of the associated Hankel matrix is computed using modular methods. The computing time analysis shows that the maximum computing time function of the modular algorithm is Ο(n5 l2), where n is the degree of the polynomial and l its length.