Maximal determinants of matrices over the roots of unity
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912274285854720 |
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| author | Ponasso, Guillermo Nuñez |
| author_facet | Ponasso, Guillermo Nuñez |
| contents | We study the maximum absolute value of the determinant of matrices with entries in the set of $\ell$-th roots of unity; this is a generalization of $D$-optimal designs and Hadamard's maximal determinant problem, which involves $\pm 1$ matrices. For general values of $\ell$, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases $\ell = 3$, $\ell = 4$ is similar to the classical Hadamard maximal determinant problem for matrices with entries $\pm 1$, and many techniques can be generalized. For $\ell = 3$ we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant over $μ_3$ for all orders $n < 14$. Additionally, we survey the case $\ell = 4$ and exhibit an infinite family of maximal determinant matrices over the fourth roots of unity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11114 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Maximal determinants of matrices over the roots of unity Ponasso, Guillermo Nuñez Combinatorics We study the maximum absolute value of the determinant of matrices with entries in the set of $\ell$-th roots of unity; this is a generalization of $D$-optimal designs and Hadamard's maximal determinant problem, which involves $\pm 1$ matrices. For general values of $\ell$, we give sharpened determinantal upper bounds and constructions of matrices of large determinant. The maximal determinant problem in the cases $\ell = 3$, $\ell = 4$ is similar to the classical Hadamard maximal determinant problem for matrices with entries $\pm 1$, and many techniques can be generalized. For $\ell = 3$ we give an additional construction of matrices with large determinant, and calculate the value of the maximal determinant over $μ_3$ for all orders $n < 14$. Additionally, we survey the case $\ell = 4$ and exhibit an infinite family of maximal determinant matrices over the fourth roots of unity. |
| title | Maximal determinants of matrices over the roots of unity |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.11114 |