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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Kapranov rank vs. tropical rank


Authors: K. H. Kim and F. W. Roush
Journal: Proc. Amer. Math. Soc. 134 (2006), 2487-2494
MSC (2000): Primary 15A99, 16Y60
Published electronically: February 8, 2006
MathSciNet review: 2213725
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Abstract | References | Similar Articles | Additional Information

Abstract: We show that determining Kapranov rank of tropical matrices is not only NP-hard over any infinite field, but if solving Diophantine equations over the rational numbers is undecidable, then determining Kapranov rank over the rational numbers is also undecidable. We prove that Kapranov rank of tropical matrices is not bounded in terms of tropical rank, answering a question of Develin, Santos, and Sturmfels.


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Additional Information

K. H. Kim
Affiliation: Department of Mathematics, Alabama State University, Montgomery, Alabama 36101-0271 – and – Fellow, Korean Academy of Science and Technology
Email: khkim@alasu.edu

F. W. Roush
Affiliation: Department of Mathematics, Alabama State University, Montgomery, Alabama 36101-0271
Email: froush@alasu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08426-7
PII: S 0002-9939(06)08426-7
Keywords: Kapranov rank, tropical rank
Received by editor(s): March 15, 2005
Published electronically: February 8, 2006
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2006 American Mathematical Society