Project : galaad
Section: New Results
Computation of normal forms in a quotient algebra
We develop a new method for computing the normal form of a polynomial modulo a zero-dimensional ideal I. We give a detailed description of the algorithm, a proof of its correctness, and finally experimentations on classical benchmark polynomial systems. The method we propose can be thought as an extension of both the Gröbner basis method and the Macaulay construction. As such it establishes a natural link between these two methods. We have weaken the monomial ordering requirement for Gröbner bases computations, which allows us to construct new type of representations for the associated quotient algebra. This approach yields more freedom in the linear algebra steps involved, which allows us to take into account numerical criteria while performing the symbolic steps. This is a new feature for a symbolic algorithm, which has an important impact on the practical efficiency, as it is illustrated by the experiments. This work is submitted for publication.
Univariate polynomial solvers in Bernstein basis
In this expository work accepted for publication (see also ), we explain how the Bernstein's basis, widely used in Computer Aided Geometric Design, provides an efficient method for real root isolation, using De Casteljau's algorithm. We explain the link between this approach and more classical methods for real root isolation such as Uspensky's method. We also present a new improved method for isolating real roots in the Bernstein's basis.
Multivariate polynomial solvers in Bernstein basis
We develop a new algorithm for solving a system of polynomials in Bernstein form. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis. It uses a powerful reduction strategy thanks to a univariate root finder based on Bezier clipping and Descarte's rule. The improvement of this reduction compared to the classical IPP method, is illustrated on systems with tangent solutions, which are the worst case situation of all iterative algorithms. We analyse the behavior and complexity of the method, proving a generalisation of Vincent's theorem for multivariate polynomials. We also show an application to the computation of self-intersection of rational Bezier curves, with examples taken up to degree 20 and some experiments on classical polynomial benchmark problems.
Univariate and multivariate Weierstrass-like methods
Participant : Olivier Ruatta.
The Weierstrass iteration is an iterative algorithm allowing to compute simultaneously all the roots of an algebraic system. This method was generalized to overconstrained systems in . More recently in , we used this method in order to address the problem of the approximate GCD and we shown the links of the iteration with the distance to the nearest consistent system. This method allows us to compute the common roots of a system near an inexact algebraic system. This work was presented as a poster to the conference ISSAC 04 at Santander and received the prize of ``distinguished poster''. In another way, in the univariate setting, we proposed a method derived from the Weierstrass iteration allowing to approximate simultaneously all the roots of an algebraic equation by integration of the proposed vector field.
Participant : Olivier Ruatta.
We proposed different algorithms to solve the multivariate Lagrange interpolation problem. Based on a previous work  where we introduced a generalization of the Lagrange basis, we proposed a generalization of the Newton basis in . This work has been presented at the EACA conference and has been published as an extended abstract in the proceedings of this conference. We obtained the better complexity bound for the multivariate Lagrange interpolation problem. The complexity of our algorithm is basically cubic in the number of interpolation points and quadratic in the complexity of the evaluation of some multivariate polynomials.
Irreducibility of multivariate subresultants
Classical subresultants of two univariate polynomials have been studied by Sylvester. Multivariate subresultants, introduced by Chardin , provide a criterion for over-constrained polynomial systems to have Hilbert function of prescribed value, generalizing the classical case. They have been used in computational algebra for polynomial system solving as well as for providing explicit formulas for the representation of rational functions. The study of their properties is an active research area, in particular it is important to know which of them are irreducible. In this work, published in , we proved the following result: let P1, ..., Pn be generic homogeneous polynomials in n variables of degrees d1, ..., dn respectively. We prove that if is an integer satisfying then all multivariate subresultants associated to the family P1, ..., Pn in degree are irreducible. We show that the lower bound is sharp. As a byproduct, we get a formula for computing the residual resultant of smooth isolated points in
Properness and inversion problems of parameterized surfaces
Rational surfaces play an important role in the frame of practical applications, especially in Computer Aided Geometric Design. Such surfaces can be parameterized, i.e. can be seen as the image of a generically finite rational map. In this collaboration, we addressed the following questions: decide if the parameterization map is invertible, and if it is the case find an inverse. Both questions have been already solved theoretically and algorithmically by means of Gröbner bases. Our approach is based on the matrix formulation of some projection operators allowing to eliminate variables. We give a general method, generalizing preliminar results obtained and presented at the international conferences EACA and ISSAC 2004 , for solving both problems by means of matrices. In particular we introduce the notion of implicitization matrices; these matrices can be used to solve simultaneously both problems by extracting information from a square matrix whose determinant is an implicit equation (that we do not need to compute!) of the surface. This work as been submitted for publication.