Lorentzian polynomials and the incidence geometry of tropical linear spaces

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 and show that it is tropically convex. 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.