On generators of bounded ratios of minors for totally positive matrices

TL;DR

This paper introduces a method to factor all bounded ratios of minors in totally positive matrices into elementary ratios, providing a new proof of Skandera's result and a necessary condition for non-principal minors.

Contribution

It generalizes previous approaches to factor ratios of minors and establishes a new necessary condition for boundedness in non-principal minors.

Findings

Provides a new proof of Skandera's result on bounded ratios

Introduces a method to factor ratios into elementary bounded ratios

Derives a necessary condition for boundedness of non-principal minors

Abstract

We provide a method for factoring all bounded ratios of the form $$\det A(I_1|I_1')\det A(I_2|I_2')/\det A(J_1|J_1')\det A(J_2|J_2')$$ where $A$ is a totally positive matrix, into a product of more elementary ratios each of which is bounded by 1, thus giving a new proof of Skandera's result. The approach we use generalizes the one employed by Fallat et al. in their work on principal minors. We also obtain a new necessary condition for a ratio to be bounded for the case of non-principal minors.