Rosario Rubio

Multivariate Polynomial Decomposition

Abstract. In this paper, we discuss several notions of decomposition for multivariate polynomials, focussing on the relation with Lüroths theorem in field theory, and the finiteness and uniqueness of decompositions. We also present two polynomial time algorithms for decomposing (sparse) multivariate polynomials over an arbitrary field.

On Multivariate Rational Function Decomposition

In this paper we discuss several notions of decomposition for multivariate rational functions, and we present algorithms for decomposing multivariate rational functions over an arbitrary field. We also provide a very efficient method to decide if a unirational field has transcendence degree one, and in the affirmative case to compute the generator.

Unirational fields of transcendence degree one and functional decomposition

In this paper we present an algorithm to compute all unirational fields of transcendence degree one, containing a given finite set of multivariate rational functions. In particular, we provide an algorithm to decompose a multivariate rational function <i>f</i> of the form <i>f</i> = <i>g</i>(<i>h</i>), where <i>g</i> is univariate rational function and <i>h</i> a multivariate one.

D-resultant for rational functions

In this paper we introduce the D-resultant of two rational functions f(t), g(t) ∈ K(t) and show how it can be used to decide if K(f(t), g(t)) = K(t) or if K[t] C K[f(t),g(t)( and to find the singularities of the parametric algebraic curve define by X = f(t),Y = g(t). In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which has its own interest.

Polynomial parametrization of curves without affine singularities

This paper gives an algorithm for computing proper polynomial parametrizations for a particular class of curves. This class is characterized by the existence of a polynomial parametrization and by the absence of affine singularities. The algorithm requires O(n3 logn) field operations, where n is the degree of the curve.

A note on implicitization and normal parametrization of rational curves

In this paper we present a method to compute an implicitization of a rational parametrized curve in an affine space over an algebraically closed field. This method is the natural generalization of the resultant method for planar curves. For this purpose we need some normality assumptions on the parametrization of the curve. Furthermore, we provide a test to decide whether a parametrization is normal and if not, we compute a normal parametrization.

On the Dimension and the Number of Parameters of a Unirational Variety

In this paper we study the relation between the dimension of a parametric variety and the number of parameters. We present an algorithm to reparameterize a variety in order to obtain a parameterization where the number of parameters equals the dimension of the variety.