On the Column Number and Forbidden Submatrices for \(\Delta\)-Modular Matrices

An integer matrix \(\mathbf{A}\) is \(\Delta\)-modular if the determinant of each \(\operatorname{rank}(\mathbf{A}) \times \operatorname{rank}(\mathbf{A})\) submatrix of \(\mathbf{A}\) has absolute value at most \(\Delta\). The study of \(\Delta\)-modular matrices appears in the theory of integer programming, where an open conjecture is whether integer programs defined by \(\Delta\)-modular constraint matrices can be solved in polynomial time if \(\Delta\) is considered constant. The conjecture is known to hold true only when \(\Delta \in \{1,2\}\). In light of this conjecture, a natural question is to understand structural properties of \(\Delta\)-modular matrices. We consider the column number question, how many nonzero, pairwise nonparallel columns can a rank-\(r\) \(\Delta\)-modular matrix have? We prove that for each positive integer \(\Delta\) and sufficiently large integer \(r\), every rank-\(r\) \(\Delta\)-modular matrix has at most \(\binom{r+1}{2} + 80\Delta^7 \cdot r\) nonzero, pairwise nonparallel columns, which is tight up to the term \(80\Delta^7\). This is the first upper bound of the form \(\binom{r+1}{2} + f(\Delta )\cdot r\) with \(f\) a polynomial function. Underlying our results is a partial list of matrices that cannot exist in a \(\Delta\)-modular matrix. We believe this partial list may be of independent interest in future studies of \(\Delta\)-modular matrices.