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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Tropical discriminants


Authors: Alicia Dickenstein, Eva Maria Feichtner and Bernd Sturmfels
Journal: J. Amer. Math. Soc. 20 (2007), 1111-1133
MSC (2000): Primary 14M25; Secondary 52B20
Published electronically: April 23, 2007
MathSciNet review: 2328718
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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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Additional Information

Alicia Dickenstein
Affiliation: Departamento de Matemática, FCEN, Universidad de Buenos Aires, (1428) B. Aires, Argentina
Email: alidick@dm.uba.ar

Eva Maria Feichtner
Affiliation: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland
Address at time of publication: Department of Mathematics, University of Stuttgart, 70569 Stuttgart, Germany
Email: feichtne@igt.uni-stuttgart.de

Bernd Sturmfels
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720
Email: bernd@math.berkeley.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-07-00562-0
PII: S 0894-0347(07)00562-0
Keywords: Tropical geometry, dual variety, discriminant.
Received by editor(s): November 8, 2005
Published electronically: April 23, 2007
Additional Notes: The first author was partially supported by UBACYT X042, CONICET PIP 5617 and ANPCYT 17-20569, Argentina.
The second author was supported by a Research Professorship of the Swiss National Science Foundation, PP002–106403/1.
The last author was partially supported by the U.S. National Science Foundation, DMS-0456960.
Dedicated: Dedicated to the memory of Pilar Pisón Casares
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.