On the Kapranov ranks of tropical matrices

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• 作者：Yaroslav Shitov ^{yaroslav-shitov@yandex.ru}
• 关键词：15A80 ; 15A03
• 刊名：Linear Algebra and its Applications
• 出版年：2012
• 期刊代码：38_00243795
• 类别：mt
• 出版时间：1 May, 2012
• 卷：436
• 期：9
• 页码：3247-3253
• 文件大小：248 K

摘要

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鈥檚 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.