On bounded ratios of minors of totally positive matrices

Abstract

We construct several examples of bounded Laurent monomials in minors of an $n\times n$ totally positive matrix which can not be factored into a product of so called primitive bounded ratios. This disproves the conjecture about factorization of bounded ratios due to Fallat, Gekhtman, and Johnson. However, all of the found examples still satisfy the subtraction-free property also conjectured in their same work. In addition, we show that the set of all of the bounded ratios forms a polyhedral cone of dimension $\left(\begin{array}{c}2n\\ n\end{array}\right)-2n$.