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

TL;DR

This paper explores the algebraic structure of the Sinkhorn limit of positive matrices, deriving a polynomial equation for its top-left entry using computational algebra, combinatorics, and AI tools, revealing new structural insights.

Contribution

It introduces a conjectural polynomial equation characterizing the Sinkhorn limit's top-left entry, combining computational and combinatorial methods, and demonstrates the use of AI in mathematical discovery.

Findings

Derived a degree-\binom{m + n - 2}{m - 1} polynomial equation

Connected the polynomial degree to minors of a matrix

Utilized AI to identify signs in determinant formulas

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.