Gema M. Diaz-Toca

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.

Dynamic Galois Theory

Given a separable polynomial over a field, every maximal idempotent of its splitting algebra defines a representation of its splitting field. Nevertheless such an idempotent is not computable when dealing with a computable field if this field has no factorization algorithm for separable polynomials. Moreover, even when such an algorithm does exist, it is often too heavy. So we suggest to address the problem with the philosophy of lazy evaluation: make only computations needed for precise results, without trying to obtain a priori complete information about the situation. In our setting, even if the splitting field is not computable as a static object, it is always computable as a dynamic one. The Galois group has a very important role in order to understand the unavoidable ambiguity of the splitting field, and this is even more important when dealing with the splitting field as a dynamic object. So it is not astonishing that successive approximations to the Galois group (which is again a dynamic object) are a good tool for improving our computations. Our work can be seen as a Galois version of the Computer Algebra software D5 (Della Dora et al., 1985).

Minors of Bezout matrices, subresultants and the parameterization of the degree of the polynomial greatest common divisor

The main purpose of this article is to present algorithms to parameterize the degree of the greatest common divisor of two polynomials with parametric coefficients: these algorithms are based on the fact that the principal minors of the Bezout matrices provide the principal subresultant sequence. When coefficients depend on parameters, these algorithms show a better behaviour than the classical ones. E-mail: gemadiaz@um.es E-mail: jounaidi@esstt.rnu.tr

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Following Mulmuleys lemma, this paper presents a generalization of the Moore--Penrose inverse for a matrix over an arbitrary field. This generalization yields a way to uniformly solve linear systems of equations which depend on some parameters.

Topology of 2D and 3D rational curves

In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or use the implicit equation of the curve (in the case of planar curves) or of any projection (in the case of space curves). Moreover, these algorithms have been implemented in Maple; the examples considered and the timings obtained show good performance skills.

Block LU factorization of Hankel and Bezout matrices and Euclidean algorithm

Given two polynomials, this paper is devoted to describing the natural relation between the Euclidean algorithm and the block LU factorization of the Hankel and Bezout matrices associated to such polynomials.

Bezout matrices, Subresultant polynomials and parameters

The main purpose of this paper is to analyze new determinantal expressions which define the subresultant sequence of two polynomials, when the coefficients of such polynomials depend on parameters.

Blind image deconvolution through Bezoutians

In this paper, we introduce a fast algorithm for computing the univariate GCD of several polynomials (not pairwise) based on the generalized Bezout matrix by using Barnetts method. This novel approach is devoted to presenting an algorithm that permits to solve the problem of blind image deconvolution by computing greatest common divisors (GCD) of several polynomials. Specifically, we design a specialized algorithm for computing the GCD of bivariate polynomials of blurred images which correspond to z -transforms to recover the original image. All algorithms have been implemented in Matlab and experimental results with synthetically blurred images are included to illustrate the effectiveness of our approach.

A new method to compute the singularities of offsets to rational plane curves

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.

Galois theory, splitting fields and computer algebra

Abstract We provide some algorithms for dynamically obtaining both a possible representation of the splitting field and the Galois group of a given separable polynomial from its universal decomposition algebra.