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We use tropical geometry to take a fresh look at the theory of $A$-discriminants of Gelfand, Kapranov and Zelevinsky. We show that the tropical $A$-discriminant is the Minkowski sum of the row space of $A$ and the Bergman fan of the kernel of $A$. The latter is a well-studied object in geometric combinatorics. As our main application, we present a positive formula for the extreme monomials of the $A$-discriminant, regardless of any smoothness assumption.

This is joint work with Alicia Dickenstein and Bernd Sturmfels.