关于具有常数实秩矩阵的二元秩

中文摘要 AI 辅助

我们继续了Parnas和Shraibman~\cite{PARNAS2026264}发起的研究，他们给出了在实数域上具有小秩的$0,1$矩阵的二元秩的上界。我们给出了在~\cite{PARNAS2026264}中借助计算机程序证明的结果的替代性纯数学证明，并且解决了其中提出的一个开放问题，即关于实秩为$5$的矩阵的最大二元秩。此外，我们的技术提供了一种通用方法，用于给出具有常数实秩矩阵的最大二元秩的非平凡上界。我们的结果也蕴含了等价问题的界，即划分一个二部图的边所需的最少双团数量，其中该二部图的约化邻接矩阵的实秩至多为$d$。

英文摘要

We continue the study initiated by Parnas and Shraibman~\cite{PARNAS2026264} who gave upper bounds on the binary rank of $0,1$ matrices which have a small rank over the reals. We give alternative completely mathematical proofs of results proved in~\cite{PARNAS2026264} with the assistance of a computer program, and also solve one of the open problems presented there regarding the maximal binary rank of a matrix with real rank $5$. Moreover, our techniques provide a general method for giving non-trivial upper bounds on the maximal binary rank of a matrix with constant real rank. Our results also imply bounds on the equivalent problem of finding the minimum number of bicliques needed to partition the edges of a bipartite graph whose reduced adjacency matrix has real rank at most $d$.

发表机构

• The Academic College of Tel Aviv-Yaffo（特拉维夫雅福学术学院）

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