{"ID":2870594,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.13463","arxiv_id":"2509.13463","title":"The column number for 3-modular matrices","abstract":"An integer-valued matrix $\\mathbf{A}$ is $Δ$-modular if each $\\text{rank}(\\mathbf{A}) \\times \\text{rank}(\\mathbf{A})$ submatrix has determinant at most $Δ$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $Δ$-modular matrix. Exact values for the column number are only known for $r \\le 2$ or $Δ\\le 2$. We prove that if $r$ is sufficiently large, then the maximum number of pairwise non-parallel columns of a rank-$r$, $3$-modular matrix is $\\binom{r+1}{2} + 2(r-1)$. This settles a conjecture by Lee, Paat, Stallknecht, and Xu on the column number in the case $Δ= 3$. We complement this main result by showing that there are at least three $3$-modular matrices with pairwise non-isomorphic vector matroids that attain this upper bound. More generally, we show that if $r \u003e Δ$, then the number of $Δ$-modular matrices with $\\binom{r+1}{2} + (Δ-1)(r-1)$ pairwise non-parallel columns and pairwise non-isomorphic vector matroids is at least exponential in $\\sqrtΔ$; previously only one matrix was known due to Lee et al.","short_abstract":"An integer-valued matrix $\\mathbf{A}$ is $Δ$-modular if each $\\text{rank}(\\mathbf{A}) \\times \\text{rank}(\\mathbf{A})$ submatrix has determinant at most $Δ$ in absolute value. The column number problem is to determine the maximum number of pairwise non-parallel columns of a rank-$r$, $Δ$-modular matrix. Exact values for...","url_abs":"https://arxiv.org/abs/2509.13463","url_pdf":"https://arxiv.org/pdf/2509.13463v1","authors":"[\"Joseph Paat\",\"Zach Walsh\",\"Luze Xu\"]","published":"2025-09-16T19:01:16Z","proceeding":"math.CO","tasks":"[\"math.CO\",\"math.OC\"]","methods":"[]","has_code":false}