$2$-Modular Matrices

Abstract

A rank-$r$ integer matrix $A$ is $Δ$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $Δ$. 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 ${r + 2 \choose 2} - 2$.

Figures (1)

• Figure 1: $A_r$ and $A'_r$.

Theorems & Definitions (52)

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