Equal Entries in Totally Positive Matrices

Abstract

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\text{-by-}n$ matrix is $\Theta (n^{4/3})$ (resp. $\Theta (n^{3/2}$)). Relationships with point-line incidences in the plane, Bruhat order of permutations, and $TP$ completability are also presented. We also examine the number and positionings of equal $2\text{-by-}2$ minors in a $2\text{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\text{-by-}2$ minors and outerplanar graphs.

15 pages