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arXiv · 2409.02789

The entries of the Sinkhorn limit of an $m \times n$ matrix

Eric Rowland·Jason Wu
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Abstract

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 Gr\"obner 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.

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Eric Rowland, Jason Wu. 2024-09-04. The entries of the Sinkhorn limit of an $m \times n$ matrix. https://arxiv.org/abs/2409.02789

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