Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

翻译：受近期关于拟阵深度度量与块结构整数规划之间关联的发现（ICALP 2020, 2022）的启发，我们对矩阵与拟阵的递归深度参数开展了系统性研究，旨在统一近年来涌现的零散概念。我们提出一个通用框架，该框架自然衍生出拟阵的八种不同深度度量，证明了它们的基本性质与相互关系，并将其与该领域的两个已有概念——拟阵分支深度及新引入的拟阵树宽的自然深度对应物——建立联系。特别地，我们证明其中六种度量两两之间功能不等价，而在这六种中，一种功能等价于拟阵分支深度，另一种等价于拟阵树深度。重要的是，我们还证明了这些深度度量在拟阵及任意域上的矩阵中保持一致，这（某种程度上令人意外）并非一个平凡的结论。最后，我们给出了拟阵参数与图经典深度度量之间的比较。