Laureano Gonzalez-Vega

Efficient topology determination of implicitly defined algebraic plane curves

This paper is devoted to present a new algorithm computing in a very efficient way the topology of a real algebraic plane curve defined implicitly. This algorithm proceeds in a seminumerical way by performing a symbolic preprocessing which allows later to accomplish the numerical computations in a very accurate way.

An Improved Upper Complexity Bound for the Topology Computation of a Real Algebraic Plane Curve

The computation of the topological shape of a real algebraic plane curve is usually driven by the study of the behavior of the curve around its critical points (which includes also the singular points). In this paper we present a new algorithm computing the topological shape of a real algebraic plane curve whose complexity is better than the best algorithms known. This is due to the avoiding, through a sufficiently good change of coordinates, of real root computations on polynomials with coefficients in a simple real algebraic extension of Q to deal with the critical points of the considered curve. In fact, one of the main features of this algorithm is that its complexity is dominated by the characterization of the real roots of the discriminant of the polynomial defining the considered curve.

Sturm—Habicht Sequences, Determinants and Real Roots of Univariate Polynomials

The real root counting problem is one of the main computational problems in Real Algebraic Geometry. It is the following: Let \(\mathbb{D}\) be an ordered domain and \(\mathbb{B}\) a real closed field containing \(\mathbb{D}\). Find algorithms which for every P ∈ \(\mathbb{D}\) [x] compute the number of roots of P in \(\mathbb{B}\). More precisely we shall study the following problem.

Implicitization of parametric curves and surfaces by using multidimensional Newton formulae

Abstract This paper is devoted to present new algorithms computing the implicit equation of a parametric plane curve and several classes of parametric surfaces in the three dimensional euclidean space. These algorithms do not require the computation of any symbolic determinant or Grobner Basis, being these tools replaced by the computation of some symmetric functions, in particular the Newton Sums, on the solution set of a well precised zero dimensional ideal.

Symbolic Recipes for Real Solutions

The main purpose of this chapter is to show how to use algorithms and methodology provided by computer algebra to manipulate in a symbolic way the real solutions of an algebraic system of equations.

Symbolic Recipes for Polynomial System Solving

In many branches of science and engineering where mathematics is used, the resolution of a problem coming from practice is often reduced to the search of a solution for a system of (algebraic or differential) equations modelling the considered problem. From our point of view, to solve a polynomial system of equations is to rewrite it (i.e., to present it in a different form) in such a way that some ‘nontrivial’ information about its solutions can be derived from this new presentation. The information mentioned above can be related to the existence or non-existence of complex or real solutions, to the number of real or complex solutions, to the approximation of one or several solutions, etc.

Algorithms in Algebraic Geometry and Applications

The present volume contains a selection of refereed papers from the MEGA-94 symposium held in Santander, Spain, in April 1994. They cover recent developments in the theory and practice of computation in algebraic geometry and present new applications in science and engineering, particularly computer vision and theory of robotics. The volume will be of interest to researchers working in the areas of computer algebra and symbolic computation as well as to mathematicians and computer scientists interested in gaining access to these topics.

Barnett's Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

This article provides a new presentation of Barnett?s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated in a companion matrix as in the original Barnett?s presentation. Moreover, this presentation also allows us to compute the coefficients of the considered greatest common divisor in an easier way than in the original Barnett?s theorems.

Parameterizing surfaces with certain special support functions, including offsets of quadrics and rationally supported surfaces

We discuss rational parameterizations of surfaces whose support functions are rational functions of the coordinates specifying the normal vector and of a given non-degenerate quadratic form. The class of these surfaces is closed under offsetting. It comprises surfaces with rational support functions and non-developable quadric surfaces, and it is a subset of the class of rational surfaces with rational offset surfaces. We show that a particular parameterization algorithm for del Pezzo surfaces can be used to construct rational parameterizations of these surfaces. If the quadratic form is diagonalized and has rational coefficients, then the resulting parameterizations are almost always described by rational functions with rational coefficients.

Applying quantifier elimination to the Birkhoff interpolation problem

Abstract This paper is devoted to show how to use Computer Algebra and Quantifier Elimination to solve some particular instances of the Birkhoff Interpolation Problem. In particular, this problem is completely solved for degrees less than or equal to 3 and any number of nodes and several instances of degree 4 and 5 by computing all the incidence normal poised matrices with such characteristics. The used Computer Algebra and Quantifier Elimination includes manipulation of multivariate polynomials, computation of determinants of matrices with polynomial entries and the formal manipulation of univariate polynomial inequalities by using Sturm—Habicht sequences and the Sign Determination Scheme.