On tropical and Kapranov ranks of tropical matrices
نویسنده
چکیده
Let M be a tropical matrix (k + x) × (k + x ′) for some k, x , x ′ ∈ N − {0} with tropical rank k. We show that Kapranov rank is k too if x and x ′ are not too big; namely if we are in one of the following cases: a) k ≥ 6 and x , x ′ ≤ 2 b) k = 4, 5, x ≤ 2 and x ′ ≤ 3 (or obviuosly the converse, that is x ≤ 3 and x ′ ≤ 2) c) k = 3 and either x , x ′ ≤ 3 or x ≤ 2 and x ′ ≤ 4 (or obviuosly the converse). This answers, negatively, to Develin, Santos and Sturmfels’question in [DSS] whether there exists a matrix 5× 5 with tropical rank 3 and Kapranov rank strictly greater than 3.
منابع مشابه
5 Kapranov Rank Vs . Tropical Rank
We show that determining Kapranov rank of tropical matrices is not only NP-hard over any field but if Diophantine equations over the rational numbers is undecidable, 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 [4].
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