%0 Generic %1 DevelinSantosSturmfels2003 %A Develin, M. %A Santos, F. %A Sturmfels, B. %D 2003 %K Barvinok Develin Kapranov Santos Sturmfels algebra geometry line matrix matroid optimization polyhedron rank semiring tropics %T On the rank of a tropical matrix %U http://arxiv.org/abs/math/0312114 %X This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed.

@misc{DevelinSantosSturmfels2003, abstract = { This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed. }, added-at = {2008-12-23T15:54:00.000+0100}, author = {Develin, M. and Santos, F. and Sturmfels, B.}, biburl = {https://www.bibsonomy.org/bibtex/272a68b15634af80fbf4aa27a246ca70d/fwd13}, description = {On the rank of a tropical matrix}, howpublished = {In "Discrete and Computational Geometry" (E. Goodman, J. Pach and E. Welzl, eds), MSRI Publications, Cambridge Univ. Press, 2005. ISBN-10: 0521848628}, interhash = {101a4f2bde74a3bd1debe217ba65129e}, intrahash = {72a68b15634af80fbf4aa27a246ca70d}, keywords = {Barvinok Develin Kapranov Santos Sturmfels algebra geometry line matrix matroid optimization polyhedron rank semiring tropics}, note = {cite arxiv:math/0312114 Comment: 20 pages, 1 figure}, timestamp = {2008-12-23T19:28:07.000+0100}, title = {On the rank of a tropical matrix}, url = {http://arxiv.org/abs/math/0312114}, year = 2003 }