Lorentzian polynomials and the incidence geometry of tropical linear spaces

TL;DR

This paper introduces Lorentzian proper position to characterize tropical linear spaces and explores their incidence geometry, revealing both parallels and differences with classical linear geometry, including structural results and properties of matroids.

Contribution

It develops a new Lorentzian framework for tropical linear spaces, providing novel structural insights and incidence properties, and introduces the concept of adjoints for tropical linear spaces.

Findings

The poset of matroids by quotient is not submodular for n ≥ 8.

Certain classical incidence properties hold for tropical linear spaces with adjoints.

New structural results on the moduli space of tropical linear subspaces.

Abstract

We introduce a notion of Lorentzian proper position in close analogy to proper position of stable polynomials. Using this notion, we give a new characterization of elementary quotients of M-convex function that parallels the Lorentzian characterization of M-convex functions. We thereby use Lorentzian proper position to study the incidence geometry of tropical linear spaces, and vice versa. In particular, we prove new structural results on the moduli space of codimension-1 tropical linear subspaces of a given tropical linear space. Applying these results, we show that some properties of classical linear incidence geometry fail for tropical linear spaces. For instance, we show that the poset of all matroids on $[n]$, partially ordered by matroid quotient, is not submodular when $n\geq 8$. On the other hand, we introduce a notion of adjoints for tropical linear spaces, generalizing adjoints of matroids, and show that certain incidence properties expected from classical geometry hold for tropical linear spaces that have adjoints.