Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Tropical Discriminants by Bernd STURMFELS, University of California at Berkeley Abstract: Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gel'fand, Kapranov and Zelevinsky. The tropical A-discriminant, which is the tropicalization of the dual variety of a toric variety, coincides with the Minkowski sum of the row space of underlying matrix A and of the tropicalization of the kernel of A. This leads to a positive formula for the extreme monomials of any A-discriminant, regardless of how singular the toric variety is. Joint work with Alicia Dickenstein and Eva Maria Feichtner.