Tropical geometry is used to develop a new approach to the theory of discriminants and resultants in the sense of Gelfand, Kapranov and Zelevinsky. The tropical -discriminant is the tropicalization of the dual variety of the projective toric variety given by an integer matrix . This tropical algebraic variety is shown to coincide with the Minkowski sum of the row space of and the tropicalization of the kernel of . This leads to an explicit positive formula for all the extreme monomials of any -discriminant.
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