拟阵伴随与零-非零矩阵模式的最小秩
Matroid adjoints and the minimum rank of zero-nonzero matrix patterns
AI总结：
本文将零-非零模式的最小秩问题推广至拟阵，通过定义基本模式与拟阵伴随，揭示了三角形数与拟阵最小秩差距的对偶关系，并证明了最小秩拟阵的唯一性及其在不同域上对可表示性的依赖。
摘要：
寻找具有给定零-非零模式的矩阵的最小秩问题已被推广到与该模式相关联的一类拟阵中。被称为三角形数的基本下界在这一推广设定中仍然成立。但是，模式的拟阵最小秩不一定与其转置的拟阵最小秩相匹配。\n我们为每个模式 X 关联一个格 L(X)。我们将拟阵 M 的基本模式定义为其超平面-点关联模式的补集，并注意到当 X 是 M 的基本模式时，M 的平坦格即为 L(X)。然后我们证明，对于每个模式 X，L(X) 的对偶格与 L(X^T) 同构。\n我们证明，与 M 具有相同秩的拟阵 M' 是 M 的伴随，当且仅当 M' 与 M 的基本模式的转置相关联。我们的主要结果将拟阵伴随的概念与三角形数 k 和模式的拟阵最小秩之间的差距现象联系在一起。具体而言，我们证明，如果与模式相关联的任何秩为 k 的拟阵具有伴随，那么该模式的转置就不存在这样的差距。\n我们证明与基本模式相关联的最小秩拟阵是唯一的。利用这一点，我们证明拟阵在不同域上的基本模式的矩阵最小秩取决于该拟阵在这些域上的可表示性。这使我们能够恢复并改进 Berman 等人的一项构造。我们还给出了一个比以往已知的任何例子都更小的模式示例，其拟阵最小秩小于它在每个域上的矩阵最小秩。最后，我们确立，对于基本模式，我们主要结果的逆命题成立。特别地，具有基本模式 X 的拟阵具有伴随，当且仅当 X^T 的拟阵最小秩等于其三角形数。
英文摘要：
The problem of finding the minimum rank of a matrix with a given zero-nonzero pattern has been generalized to a class of matroids associated to the pattern. The fundamental lower bound known as the triangle number still holds in this generalized setting. But the matroid minimum rank of a pattern need not match that of its transpose. We associate to each pattern $X$ a lattice $L(X)$. We define the fundamental pattern of a matroid $M$ to be the complement of its hyperplane-point incidence pattern and note that when $X$ is the fundamental pattern of $M$, the lattice of flats of $M$ is $L(X)$. We then prove that, for every pattern $X$, the dual lattice of $L(X)$ is isomorphic to $L(X^T)$. We show that a matroid $M'$ of the same rank as $M$ is an adjoint of $M$ if and only if $M'$ is associated with the transpose of the fundamental pattern of $M$. Our main result ties together the notion of a matroid adjoint with the phenomenon of a gap between the triangle number $k$ and the matroid minimum rank of a pattern. Namely, we show that, if any matroid of rank $k$ associated with a pattern has an adjoint, then there is no such gap for the pattern's transpose. We show that the matroid of minimum rank associated with the fundamental pattern is unique. Using this, we prove that the matrix minimum rank of the fundamental pattern of a matroid over different fields depends on the representability of the matroid over those fields. This allows us to recover and improve upon a construction of Berman et al. We also give a smaller example than any previously known of a pattern with a matroid minimum rank smaller than its matrix minimum rank over every field. Finally, we establish that, for the fundamental pattern, a converse holds to our main result. In particular, a matroid with fundamental pattern $X$ has an adjoint if and only if the matroid minimum rank of $X^T$ is equal to its triangle number.
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