$2$-Modular Matrices

TL;DR

This paper investigates the maximum number of nonzero, pairwise non-parallel rows in rank-$r$ 2-modular matrices, extending classical results from unimodular matrices to a broader class.

Contribution

It establishes a new upper bound for the number of such rows in rank-$r$ 2-modular matrices for large $r$, generalizing Heller's 1957 result.

Findings

Maximum number of rows in rank-$r$ 2-modular matrices is ${r + 2 race 2} - 2$ for large $r$.

Extends classical unimodular matrix results to 2-modular matrices.

Provides insights into the structure of $oxed{ ext{2-modular matrices}}$.

Abstract

A rank-$r$ integer matrix $A$ is $\Delta$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $\Delta$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer programming theory and matroid theory. A 1957 result of Heller shows that the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ unimodular matrix is ${r + 1 \choose 2}$. We prove that, for each sufficiently large integer $r$, the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ $2$-modular matrix is ${r + 2 \choose 2} - 2$.