Section: New Results

Algebra

Solving algebraic equations

Participant : Bernard Mourrain.

Algebraic methods to solve polynomial equations have been described in the tutorial presentation [14] . In the first one, we give an overview of fundamental algebraic properties, used to recover the roots of a polynomial system, when we know the multiplicative structure of its quotient algebra. This involves ingredients such as tables of multiplication, duality, eigenvector computations, Chow form. In a seccond tutorial to be published we describe normal form computation and more precisely border basis methods, that we illustrate several simple examples. In particular, we show its usefulness in the context of solving polynomial systems with approximate coefficients. The main results are recalled and we prove a new result on the syzygies, naturally associated with commutation properties. Finally, we describe an algorithm and its implementation for computing such border bases.

Univariate polynomial solvers

Participants : Ioannis Emiris [ Univ. Athens ] , Bernard Mourrain, Elias Tsigaridas [ Univ. Athens ] .

In [24] , we present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree d, with integer coefficients of bit size , using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of . This matches the best known bounds for binary subdivision solvers. Moreover, we generalize this to cover the non square-free polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities, and we improve the known bounds at least by a factor of d. Finally, we present our C++ implementation in synaps and some experiments on various data sets. This work has been accepted for publication.

Implicitization of rational hypersurfaces

Participants : Laurent Busé, Marc Chardin [ Univ. Paris VI ] , Jean-Pierre Jouanolou [ Univ. Strasbourg ] .

Recently, a method to compute the implicit equation of a parameterized hypersurface has been developed by the authors. We address here some questions related to this method: optimality of a degree estimate and determination of an extraneous factor that appears when the base points are not locally complete intersections. We then make a link with a resultant computation for the case of rational plane curves and for particular cases of space surfaces. As a consequence, we prove a conjecture of Busé, Cox and D'Andrea. This work has been submitted for publication.

Factors of iterated resultants

Participants : Laurent Busé, Bernard Mourrain.

In this work we were interested in the understanding of the factorization of iterated resultants and discriminants that we encounter for instance in the study of the topology of an algebraic surface. Our contribution was to give all the irreducible factors of two times iterated univariate resultants and discriminants over the integer universal ring of coefficients of the entry polynomials. Moreover, we showed that each factor can be separately and explicitely computed in terms of a certain multivariate resultant. A paper [23] has been submitted for publication.

Discriminants of homogeneous polynomials

Participants : Laurent Busé, Jean-Pierre Jouanolou [ Univ. Strasbourg ] .

This is the continuation of a work, still in progress, started last year which aims to provide a computation study of discriminants of homogeneous polynomials. These objects give a necessary and sufficient condition so that a given variety in a projective space has singularities; they are thus of interest in CAGD since it is useful to detect singularities of a curve or algebraic surfaces which are supposed to represent real objects. This work is done in collboration with Jean-Pierre Jouanolou from the university Louis Pasteur of Strasbourg.

Residue Calculus and Applications

Participants : Mohamed Elkadi, Alain Yger [ Univ. Bordeaux ] .

We present a new algorithm to compute effectively the residue of a polynomial maps inspired by the so-called Arnold's perturbation method. The strategy is to reduce the computations to the case of a map without common zeroes at infinity, and to deduce algebraic relations between the components of this map and the coordinate functions. Then we use a generalized version of the transformation to obtain the residue. Instead of computing symbolic determinants, we rely on constant coefficients of some Laurent series. The algorithm is then applied to solve some questions in Computer Aided Geometric Design.