矩阵马尔可夫复杂性的渐近性
Asymptotic behavior of Markov complexity of matrices
摘要：
对于任何整数矩阵$A$，可以将由图和另一个整数矩阵$A_B$组成的拟阵结构关联起来。这个图的连通分量称为花束。我们证明了花束表现良好的$r$ -th劳伦斯提升矩阵，我们用它来证明，马尔可夫和格雷弗复杂性的$m\times n$矩阵秩$d$可能是任意大的$n\geq 4$和$d\leq n-2$。相反，我们证明了它们是有界的$n$和最大的绝对值$a$的任何条目的$A$。
英文摘要：
To any integer matrix $A$ one can associate a matroid structure consisting of a graph and another integer matrix $A_B$. The connected components of this graph are called bouquets. We prove that bouquets behave well with respect to the $r$--th Lawrence liftings of matrices and we use it to prove that the Markov and Graver complexities of $m\times n$ matrices of rank $d$ may be arbitrarily large for $n\geq 4$ and $d\leq n-2$. In contrast, we show they are bounded in terms of $n$ and the largest absolute value $a$ of any entry of $A$.
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