Lorentzian polynomials and the incidence geometry of tropical linear spaces

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.

💡 Research Summary

The paper introduces “Lorentzian proper position,” a notion modeled on the proper‑position relation for stable polynomials, and uses it to bridge Lorentzian polynomial theory with the combinatorics of M‑convex functions and valuated matroids. For two non‑zero Lorentzian polynomials f (degree d) and g (degree d + 1), the relation f ≪ₗ g holds when the sum g + wₙ₊₁ f is again Lorentzian. This mirrors the stable‑polynomial condition f ≪ g, but the Lorentzian version is only partially convex: the set {g | f ≪ₗ g}∪{0} is a closed convex cone, whereas the dual set {h | h ≪ₗ f} need not be convex.

The authors translate elementary quotients of valuated matroids into this language. Given a valuated matroid µ of rank d and an elementary quotient θ of rank d − 1, they associate to each a Lorentzian polynomial f_{µ,q} and f_{θ,q} (with a parameter 0 < q ≤ 1). Theorem A proves that µ ↠ θ (θ is a quotient of µ) if and only if f_{θ,q} ≪ₗ f_{µ,q} for every q. This recovers a result of Fink‑Moci in a Lorentzian framework and provides a new characterization of elementary quotients of M‑convex functions.

The paper then studies the “relative Dressian” Dr₁(µ), the set of all codimension‑1 tropical linear subspaces of the tropical linear space Trop µ. Corollary 1.4 (originally due to Fink‑Moci) shows that Dr₁(µ) is tropically convex: any tropical linear combination of two codimension‑1 subspaces remains a codimension‑1 subspace of Trop µ. Theorem 4.3 strengthens this by describing Dr₁(µ) as a tropical prevariety cut out by a concrete collection of tropical hyperplanes, and identifies it with the order complex of the elementary quotient lattice cQt₁(µ). When µ is uniform, this recovers earlier work of Ardila‑Klivans‑Williams.

Theorem B investigates the converse convexity direction. For each q, the cone generated by {h | h ≪ₗ f_{µ,q}}∪{0} contains the convex cone spanned by the polynomials f_{θ,q} with