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

Abstract

We use a variety of computational tools to obtain a degree-$\left(\genfrac{}{}{0ex}{}{m+n-2}{m-1}\right)$ 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.

27 pages; Mathematica package and documentation available as ancillary files; this version includes the signs of the off-diagonal entries, giving a complete equation