Abstract. An integer matrix [Formula: see text] is [Formula: see text]-modular if the determinant of each [Formula: see text] submatrix of [Formula: see text] has absolute value at most [Formula: see text]. The study of [Formula: see text]-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by [Formula: see text]-modular constraint matrices can be solved in polynomial time if [Formula: see text] is considered constant. The conjecture is known to hold true only when [Formula: see text]. In light of this conjecture, a natural question is to understand structural properties of [Formula: see text]-modular matrices. We consider the column number question, how many nonzero, pairwise nonparallel columns can a rank-[Formula: see text] [Formula: see text]-modular matrix have? We prove that for each positive integer [Formula: see text] and sufficiently large integer [Formula: see text], every rank-[Formula: see text] [Formula: see text]-modular matrix has at most [Formula: see text] nonzero, pairwise nonparallel columns, which is tight up to the term [Formula: see text]. This is the first upper bound of the form [Formula: see text] with [Formula: see text] a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a [Formula: see text]-modular matrix. We believe this partial list may be of independent interest in future studies of [Formula: see text]-modular matrices.

An integer matrix $A$ is $\Delta$-modular if the determinant of each $rank(A) \times rank(A)$ 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 columns of a rank-$r$ unimodular matrix is ($r + 1 \atop 2$). We prove that, for each sufficiently large integer $r$, the maximum number of nonzero, pairwise non-parallel columns of a rank-$r$ 2-modular matrix is ($r + 2 \atop 2$)$ - 2$.

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific nonbinary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit.