AI 中文总结
该研究针对Δ-模矩阵，基于Sauer矩阵开展精细分析，将其不同列数的上界从O(m⁴Δ)优化至O(m³Δ)，完善了相关理论界的估计。
AI 中文摘要
本文在Gennadiy Averkov与Matthias Schymura（2022）启动的分析基础上，证明秩为m的Δ-模矩阵A∈ℤ^(m×n)的不同列数为O(m³Δ)，从而改进了此前O(m⁴Δ)的界。需注意，若一个矩阵的所有m阶子式的最大绝对值恰好为Δ，则称其为Δ-模矩阵。
英文摘要
In this paper, we build upon the analysis initiated by Gennadiy Averkov and Matthias Schymura (2022) and establish that the number of distinct columns of a $Δ$-modular matrix $A \in \mathbb{Z}^{m \times n}$ of rank $m$ is $O(m^3 Δ)$. This upper bound was previously known only for odd values of $Δ$. Recall that a matrix is called $Δ$-modular if the maximum of the absolute values of its $m \times m$ minors equals $Δ$.
