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Dive into the research topics where Laurent Busé is active.

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Featured researches published by Laurent Busé.


Journal of Algebra and Its Applications | 2003

IMPLICITIZATION OF SURFACES IN ℙ3 IN THE PRESENCE OF BASE POINTS

Laurent Busé; David Cox; Carlos D'Andrea

We show that the method of moving quadrics for implicitizing surfaces in ℙ3 applies in certain cases where base points are present. However, if the ideal defined by the parametrization is saturated, then this method rarely applies. Instead, we show that when the base points are a local complete intersection, the implicit equation can be computed as the resultant of the first syzygies.


Journal of Symbolic Computation | 2005

Implicitizing rational hypersurfaces using approximation complexes

Laurent Busé; Marc Chardin

We describe an algorithm for implicitizing rational hypersurfaces with at most a finite number of base points, based on a technique described in Buse, Laurent, Jouanolou, Jean-Pierre [2003. On the closed image of a rational map and the implicitization problem. J. Algebra 265, 312-357], where implicit equations are obtained as determinants of certain graded parts of an approximation complex. We detail and improve this method by providing an in-depth study of the cohomology of such a complex. In both particular cases of interest of curve and surface implicitization we also present algorithms which involve only linear algebra routines.


Journal of Symbolic Computation | 2000

Generalized resultants over unirational algebraic varieties

Laurent Busé; Mohamed Elkadi; Bernard Mourrain

In this paper, we propose a new method, based on Bezoutian matrices, for computing a nontrivial multiple of the resultant over a projective variety X, which is described on an open subset by a parameterization. This construction, which generalizes the classical and toric one, also applies for instance to blowing up varieties and to residual intersection problems. We recall the classical notion of resultant over a variety X. Then we extend it to varieties which are parameterized on a dense open subset and give new conditions for the existence of the resultant over these varieties. We prove that any maximal nonzero minor of the corresponding Bezoutian matrix yields a nontrivial multiple of the resultant. We end with some experiments.


computer algebra in scientific computing | 2005

Resultant-based methods for plane curves intersection problems

Laurent Busé; Houssam Khalil; Bernard Mourrain

We present an algorithm for solving polynomial equations, which uses generalized eigenvalues and eigenvectors of resultant matrices. We give special attention to the case of two bivariate polynomials and the Sylvester or Bezout resultant constructions. We propose a new method to treat multiple roots, detail its numerical aspects and describe experiments on tangential problems, which show the efficiency of the approach. An industrial application of the method is presented at the end of the paper. It consists in recovering cylinders from a large cloud of points and requires intensive resolution of polynomial equations.


Mathematics of Computation | 2009

EXPLICIT FACTORS OF SOME ITERATED RESULTANTS AND DISCRIMINANTS

Laurent Busé; Bernard Mourrain

In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into irreducible factors of several constructions involving two times iterated univariate resultants and discriminants over the integer universal ring of coefficients of the entry polynomials. Cases involving from two to four generic polynomials and resultants or discriminants in one of their variables are treated. The decompositions into irreducible factors we get are obtained by exploiting fundamental properties of the univariate resultants and discriminants and induction on the degree of the polynomials. As a consequence, each irreducible factor can be separately and explicitly computed in terms of a certain multivariate resultant. With this approach, we also obtain as direct corollaries some results conjectured by Collins and McCallum which correspond to the case of polynomials whose coefficients are themselves generic polynomials in other variables. Finally, a geometric interpretation of the algebraic factorization of the iterated discriminant of a single polynomial is detailled.


arXiv: Commutative Algebra | 2009

Torsion of the symmetric algebra and implicitization

Laurent Busé; Marc Chardin; Jean Pierre Jouanolou

Recently, a method to compute the implicit equation of a parametrized hypersurface has been developed by the authors. We address here some questions related to this method. First, we prove that the degree estimate for the stabilization of the MacRaes invariant of a graded part of the symmetric algebra is optimal. Then we show that the extraneous factor that may appear in the process splits into a product a linear forms in the algebraic closure of the base field, each linear form being associated to a non complete intersection base point. Finally, we make a link between this method and a resultant computation for the case of rational plane curves and space surfaces.


international symposium on symbolic and algebraic computation | 2001

Residual resultant over the projective plane and the implicitization problem

Laurent Busé

In this article, we first generalize the recent notion of residual resultant of a complete intersection [4] to the case of a local complete intersection of codimension 2 in the projective plane, which is the necessary and sufficient condition for a system of three polynomials to have a solution “outside” a variety, defined here by a local complete intersection of codimension 2. We give its degree in the coefficients of each polynomial and compute it as the god of three polynomials or as a product of two determinants divided by another one. In a second part we use this new type of resultant to give a new method to compute the implicit equation of a rational surface with base points in the case where these base points are a local complete intersection of codimension 2.


Journal of Pure and Applied Algebra | 2001

Resultant over the residual of a complete intersection

Laurent Busé; Mohamed Elkadi; Bernard Mourrain

In this article, we study the residual resultant which is the necessary and sufficient condition for a polynomial system F to have a solution in the residual of a variety, defined here by a complete intersection G. We show that it corresponds to an irreducible divisor and give an explicit formula for its degree in the coefficients of each polynomial. Using the resolution of the ideal (F:G) and computing its regularity, we give a method for computing the residual resultant using a matrix which involves a Macaulay and a Bezout part. In particular, we show that this resultant is the gcd of all the maximal minors of this matrix. We illustrate our approach for the residual of points and end by some explicit examples.


Computer Aided Geometric Design | 2008

Division algorithms for Bernstein polynomials

Laurent Busé; Ron Goldman

Three division algorithms are presented for univariate Bernstein polynomials: an algorithm for finding the quotient and remainder of two univariate polynomials, an algorithm for calculating the GCD of an arbitrary collection of univariate polynomials, and an algorithm for computing a @m-basis for the syzygy module of an arbitrary collection of univariate polynomials. Division algorithms for multivariate Bernstein polynomials and analogues in the multivariate Bernstein setting of Grobner bases are also discussed. All these algorithms are based on a simple ring isomorphism that converts each of these problems from the Bernstein basis to an equivalent problem in the monomial basis. This isomorphism allows all the computations to be performed using only the original Bernstein coefficients; no conversion to monomial coefficients is required.


Computer Aided Geometric Design | 2010

Matrix-based implicit representations of rational algebraic curves and applications

Laurent Busé; Thang Luu Ba

Given a parameterization of an algebraic rational curve in a projective space of arbitrary dimension, we introduce and study a new implicit representation of this curve which consists in the locus where the rank of a single matrix drops. Then, we illustrate the advantages of this representation by addressing several important problems of Computer Aided Geometric Design: the point-on-curve and inversion problems, the computation of singularities and the calculation of the intersection between two rational curves.

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André Galligo

University of Nice Sophia Antipolis

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Mohamed Elkadi

University of Nice Sophia Antipolis

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Guillaume Chèze

Institut de Mathématiques de Toulouse

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Ioannis Z. Emiris

National and Kapodistrian University of Athens

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Bernard Mourrain

Centre national de la recherche scientifique

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Michel Merle

University of Nice Sophia Antipolis

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