On the Kapranov ranks of tropical matrices |
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Authors: | Yaroslav Shitov |
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Institution: | Moscow State University, Leninskie Gory, GSP-1, Moscow 119991, Russia |
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Abstract: | We investigate the Kapranov rank functions of tropical matrices for different ground fields. For any infinite ground field we show that the rank-product inequality holds for Kapranov rank, and we prove that the Kapranov rank respects Green’s preorders on the semigroup of tropical n-by-n matrices. The rank-product inequality is shown to fail for Kapranov rank over any finite ground field. We provide an example of a 7-by-7 01-matrix whose Kapranov rank is independent of a ground field, equals 6, and exceeds tropical rank. |
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