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

DOI:
https://doi.org/10.1090/S0002-9939-06-08426-7

Published electronically:
February 8, 2006

MathSciNet review:
2213725

Full-text PDF Free Access

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:
https://doi.org/10.1090/S0002-9939-06-08426-7

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