Positroids, Dressian and stable polynomials

TL;DR

This paper explores the relationship between Lorentzian polynomials, the Dressian, and positroids, establishing new connections and confirming a conjecture for Rayleigh matroids that include positroids.

Contribution

It demonstrates that multiaffine homogeneous stable polynomials relate to positroids and confirms a conjecture for Rayleigh matroids containing positroids.

Findings

Stable polynomials relate to positroids.

Conjecture holds for Rayleigh matroids.

Further results for matroid classes.

Abstract

Our work is motivated by the connection established between Lorentzian polynomials and the Dressian in the seminal work of Br\"and\'en and Huh on Lorentzian polynomials. We analyze this relation for the class of positroids, and are able to show that in this case, we can relate a multiaffine homogenous stable polynomial to it. Additionally, we also highlight that a conjecture for matroids posed by Br\"and\'en and Huh is true when considered over the class of Rayleigh matroids which strictly contain the class of positroids. We collect these findings along with other results for further exploration.