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.