The entries of the Sinkhorn limit of an $m \times n$ matrix
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916755877658624 |
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| author | Rowland, Eric Wu, Jason |
| author_facet | Rowland, Eric Wu, Jason |
| contents | We use a variety of computational tools to obtain a degree-$\binom{m + n - 2}{m - 1}$ polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive $m \times n$ matrix. The degree of this equation has a combinatorial interpretation as the number of minors of an $(m - 1) \times (n - 1)$ matrix, and the coefficients involve a determinant formula that reflects new combinatorial structure on sets of minor specifications. The tools we use include Gröbner bases, which produce equations for small matrices; the PSLQ algorithm, which produces equations for larger matrices as part of an interpolation effort that required 1.5 years of CPU time; and ChatGPT o3-mini-high, which identified the signs of the off-diagonal entries in the determinant formula. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02789 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The entries of the Sinkhorn limit of an $m \times n$ matrix Rowland, Eric Wu, Jason Number Theory Commutative Algebra Combinatorics We use a variety of computational tools to obtain a degree-$\binom{m + n - 2}{m - 1}$ polynomial equation conjecturally satisfied by the top-left entry of the Sinkhorn limit of a positive $m \times n$ matrix. The degree of this equation has a combinatorial interpretation as the number of minors of an $(m - 1) \times (n - 1)$ matrix, and the coefficients involve a determinant formula that reflects new combinatorial structure on sets of minor specifications. The tools we use include Gröbner bases, which produce equations for small matrices; the PSLQ algorithm, which produces equations for larger matrices as part of an interpolation effort that required 1.5 years of CPU time; and ChatGPT o3-mini-high, which identified the signs of the off-diagonal entries in the determinant formula. |
| title | The entries of the Sinkhorn limit of an $m \times n$ matrix |
| topic | Number Theory Commutative Algebra Combinatorics |
| url | https://arxiv.org/abs/2409.02789 |