On generators of bounded ratios of minors for totally positive matrices

Adam Boocher; Bradley Froehle

doi:10.1016/j.laa.2007.10.011

paper

On generators of bounded ratios of minors for totally positive matrices

arXiv:0804.3226 · doi:10.1016/j.laa.2007.10.011

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.

27 pages, 2 figures