On bounded ratios of minors of totally positive matrices
Daniel Soskin, Michael Gekhtman

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 .
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Taxonomy
TopicsMatrix Theory and Algorithms · graph theory and CDMA systems · Advanced Topics in Algebra
