$2$-Modular Matrices
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866913338334642176 |
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| author | Oxley, James Walsh, Zach |
| author_facet | Oxley, James Walsh, Zach |
| contents | 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 ${r + 2 \choose 2} - 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_04525 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | $2$-Modular Matrices Oxley, James Walsh, Zach Combinatorics 05B35, 90C10 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 ${r + 2 \choose 2} - 2$. |
| title | $2$-Modular Matrices |
| topic | Combinatorics 05B35, 90C10 |
| url | https://arxiv.org/abs/2105.04525 |