Equal Entries in Totally Positive Matrices

TL;DR

This paper determines the maximum number of equal entries in totally positive matrices, linking combinatorial geometry, permutation orderings, and matrix completion properties.

Contribution

It establishes asymptotic bounds on equal entries in totally positive matrices and explores their geometric and combinatorial implications.

Findings

Maximal equal entries in TP matrices are Θ(n^{4/3})

Maximal equal entries in totally nonsingular matrices are Θ(n^{3/2})

Relationships with point-line incidences and permutation orders are identified.

Abstract

We show that the maximal number of equal entries in a totally positive (resp. totally nonsingular) $n\textrm{-by-}n$ 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 $2\textrm{-by-}2$ minors in a $2\textrm{-by-}n$ $TP$ matrix, and give a relationship between the location of equal $2\textrm{-by-}2$ minors and outerplanar graphs.