Duality in Effective Algebraic Geometry.

                     Bernard Mourrain
                   INRIA, Nice, France
            Bernard.Mourrain@sophia.inria.fr

In this presentation, we will focus on the use of duality in effective
computations on polynomials.  Its links with the theory of inverse
systems and an application to the construction of the local ring of an
isolated point will be described.  Its correlation with structured
matrices and some natural extensions to the multivariate case will be
discussed including multivariate Toeplitz, Hankel, and Van der Monde
matrices, Bezoutians, algebraic residues and relations between them.
Some applications to root finding problems for a system of
multivariate polynomial equations will be given. Finallym, we will
show how this techniques enable us to obtain a better insight into the
major problems of multivariate polynomial computations and to improve
substantially some known algorithms.