Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Computing the Multiplicity Structure via Duality by Zhonggang ZENG, Northeastern Illinois University Abstract: This talk presents a numerical algorithm for computing the multiplicity structure of a polynomial system at an isolated zero by calculating the dual space of the ideal. The algorithm identifies the multiplicity, the Hilbert function, a basis of differential functionals for the dual space along with its depth and breadth. Using numerical rank-revealing, the algorithm allows the zero to be either exact or approximate while the polynomial system can be intrinsic or empirical. As an application, the duality analysis and methodology are used to analyze deflation methods in solving singular polynomial systems and to establish deflation bounds.