On the Maximal Number of Columns of a $\Delta$-modular Integer Matrix: Bounds and Computations

TL;DR

This paper investigates the maximum number of distinct columns in a $ ext{Δ}$-modular integer matrix, providing bounds that depend on either the number of rows or the modular parameter, advancing understanding of matrix column limits.

Contribution

It introduces a new upper bound of order $O( ext{Δ})$ for fixed number of rows, complementing previous results and clarifying the relationship between matrix size and modular constraints.

Findings

Established an $O( ext{Δ})$ upper bound for fixed $m$

Extended understanding of column limits in $ ext{Δ}$-modular matrices

Provided bounds with polynomial dependence on $m$

Abstract

We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and obtain an upper bound of the form $O(\Delta)$ for fixed $m$, and with the implied constant depending polynomially on $m$.