Tropical discriminants

Dickenstein, Alicia; Feichtner, Eva; Sturmfels, Bernd

doi:10.1090/S0894-0347-07-00562-0

Tropical discriminants
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by Alicia Dickenstein, Eva Maria Feichtner and Bernd Sturmfels

J. Amer. Math. Soc. 20 (2007), 1111-1133

DOI: https://doi.org/10.1090/S0894-0347-07-00562-0

Published electronically: April 23, 2007

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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 is the tropicalization of the dual variety of the projective toric variety given by an integer matrix $A$. This tropical algebraic variety is shown to coincide with the Minkowski sum of the row space of $A$ and the tropicalization of the kernel of $A$. This leads to an explicit positive formula for all the extreme monomials of any $A$-discriminant.

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