Permanental inequalities for totally positive matrices

TL;DR

This paper develops inequalities for permanents of submatrices in totally positive matrices, extending previous minor ratio results and characterizing positive polynomial differences on these matrices.

Contribution

It provides a permanental analog of known minor ratio inequalities and characterizes positive polynomial differences on totally positive matrices.

Findings

Characterization of ratios of permanents bounded on totally positive matrices.

Extension of results to differences of monomials evaluating positively.

Analog of minor ratio inequalities for permanents.

Abstract

We characterize ratios of permanents of (generalized) submatrices which are bounded on the set of all totally positive matrices. This provides a permanental analog of results of Fallat, Gekhtman, and Johnson [{\em Adv.\ Appl.\ Math.} {\bf 30} no.\ 3, (2003) pp.\ 442--470] concerning ratios of matrix minors. We also extend work of Drake, Gerrish, and the first author [{\em Electron.\ J.\ Combin.,} {\bf 11} no.\ 1, (2004) Note 6] by characterizing the differences of monomials in $\mathbb{Z}[x_{1,1},x_{1,2},...,x_{n,n}]$ which evaluate positively on the set of all totally positive $n \times n$ matrices.