A counterexample on tropical linear spaces

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• 作者：Yaroslav Shitov ^{yaroslav-shitov@yandex.ru" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14T05
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：511
• 期：Complete
• 页码：19-21
• 全文大小：196 K

文摘

Let K be a field with non-Archimedean valuation v, and assume A   is a matrix of size m×n$m\times n$ and rank k over K. Richter-Gebert, Sturmfels, and Theobald proved that the rows of A   are a tropical basis of the corresponding linear space whenever m=k$m=k$ and any submatrix formed by k   columns of v(A)$v(A)$ has tropical rank k  . We show that this result is no longer true without assumption m=k$m=k$, refuting the conjecture proposed by Yu and Yuster in 2007.