On bounded ratios of minors of totally positive matrices

TL;DR

This paper investigates the structure of bounded ratios of minors in totally positive matrices, providing counterexamples to a previous conjecture and characterizing the geometric structure of all such ratios.

Contribution

It demonstrates that certain bounded ratios cannot be factored into primitive ratios, disproving a prior conjecture, and characterizes the set of all bounded ratios as a specific polyhedral cone.

Findings

Counterexamples to the factorization conjecture.

Bounded ratios form a polyhedral cone of specific dimension.

All found examples satisfy the subtraction-free conjecture.

Abstract

We provide several examples of bounded Laurent monomials of minors of a totally positive matrix, which can not be factored into a product of so called primitive ratios, thus showing that the conjecture about factorization of bounded ratios stated in [3] by Fallat, Gekhtman, and Johnson does not hold. However, all found examples satisfy subtraction-free conjecture stated also in [3]. In addition, we show that the set of all bounded ratios form a polyhedral cone of dimension $\binom{2n}{n}-2n$.