Abstract Suppose the only thing we know about a matrix is which of its entries are zero. What can we say about its rank? We develop a framework for applying matroid theory to this question. One consequence is a generalization of the question to the setting of matroids; over an infinite field, we recover the original question by considering only matroids representable over that field. This framework also lets us revisit prior work on the problem for matrices and clarify how previous results rest on facts about matroids. In the process, we unify and simplify the proofs of these results, and sometimes improve on them a bit. We also derive a few new results, and consider further directions for exploration.

中文翻译：

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拟阵和矩阵模式的最小秩问题

摘要 假设我们对矩阵的唯一了解是它的哪些条目为零。关于它的排名，我们能说什么？我们开发了一个将拟阵理论应用于这个问题的框架。一个结果是将问题推广到拟阵的设置；在无限域上，我们通过只考虑在该域上可表示的拟阵来恢复原始问题。这个框架还让我们重新审视先前关于矩阵问题的工作，并阐明先前的结果如何依赖于拟阵的事实。在这个过程中，我们对这些结果的证明进行了统一和简化，有时对它们进行了一些改进。我们还得出了一些新结果，并考虑了进一步的探索方向。