Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


Mixed subdivisions and ranks of tropical matrices

Author: Yaroslav Shitov
Journal: Proc. Amer. Math. Soc. 142 (2014), 15-19
MSC (2010): Primary 15A03, 15A80
Published electronically: September 6, 2013
MathSciNet review: 3119176
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We show that neither the Barvinok rank nor the Kapranov rank of a tropical matrix $ M$ can be defined in terms of the regular mixed subdivision produced by $ M$. This answers a question asked by Develin, Santos and Sturmfels.

References [Enhancements On Off] (What's this?)

  • [1] Marianne Akian, Stéphane Gaubert, and Alexander Guterman, Linear independence over tropical semirings and beyond, Tropical and idempotent mathematics, Contemp. Math., vol. 495, Amer. Math. Soc., Providence, RI, 2009, pp. 1-38. MR 2581511 (2011e:16085),
  • [2] Melody Chan, Anders Jensen, and Elena Rubei, The $ 4\times 4$ minors of a $ 5\times n$ matrix are a tropical basis, Linear Algebra Appl. 435 (2011), no. 7, 1598-1611. MR 2810658 (2012e:14123),
  • [3] Mike Develin, Francisco Santos, and Bernd Sturmfels, On the rank of a tropical matrix, Combinatorial and computational geometry, Math. Sci. Res. Inst. Publ., vol. 52, Cambridge Univ. Press, Cambridge, 2005, pp. 213-242. MR 2178322 (2006g:15001)
  • [4] Francisco Santos, The Cayley trick and triangulations of products of simplices, Integer points in polyhedra--geometry, number theory, algebra, optimization, Contemp. Math., vol. 374, Amer. Math. Soc., Providence, RI, 2005, pp. 151-177. MR 2134766 (2005k:52025),
  • [5] Ya. Shitov, Example of a $ 6$-by-$ 6$ matrix with different tropical and Kapranov ranks, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2011 (5), 58-61. MR 2964361
  • [6] Yaroslav Shitov, On the Kapranov ranks of tropical matrices, Linear Algebra Appl. 436 (2012), no. 9, 3247-3253. MR 2900712,

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A03, 15A80

Retrieve articles in all journals with MSC (2010): 15A03, 15A80

Additional Information

Yaroslav Shitov
Affiliation: Moscow State University, Leninskie Gory, 119991, GSP-1, Moscow, Russia
Address at time of publication: National Research University – Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow, 101000, Russia

Keywords: Tropical semiring, matrix theory, Barvinok rank, Kapranov rank.
Received by editor(s): October 3, 2011
Received by editor(s) in revised form: February 22, 2012
Published electronically: September 6, 2013
Communicated by: Jim Haglund
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society