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Tree metrics and log-concavity for matroids

Federico Ardila-Mantilla, Sergio Cristancho, Graham Denham, Christopher Eur, June Huh, Botong Wang

TL;DR

The paper addresses the problem of characterizing gross substitutes (M^natural-concave) set functions via Lorentzian generating polynomials. It proves that a function $\nu$ is M^natural-concave if and only if its homogeneous generating polynomial $Z_{q,\nu}$ is Lorentzian for all $0<q\le 1$, unifying discrete convexity with Lorentzian polynomial theory and resolving a longstanding question of Eur–Huh. A key analytic tool is a rank-1 bound for the leaf-distance matrix of ultrametric trees, refining Graham–Pollak’s single-positive-eigenvalue result and enabling an inductive Lorentzian argument. As consequences, the authors establish strengthened polynomial inequalities for Mason-type log-concavity in both valuated and ordinary matroids, answering open questions by Giansiracusa–Rincón–Schleis–Ulirsch and Pak. Overall, the work builds a bridge between matroid theory, discrete convex analysis, and Lorentzian polynomials, yielding new inequalities and a framework with potential impacts on combinatorial Hodge theory and matroid valuation.

Abstract

We show that a set function $ν$ satisfies the gross substitutes property if and only if its homogeneous generating polynomial $Z_{q,ν}$ is a Lorentzian polynomial for all positive $q \le 1$, answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rincón-Schleis-Ulirsch for valuated matroids, and another posed by Pak for ordinary matroids.

Tree metrics and log-concavity for matroids

TL;DR

The paper addresses the problem of characterizing gross substitutes (M^natural-concave) set functions via Lorentzian generating polynomials. It proves that a function is M^natural-concave if and only if its homogeneous generating polynomial is Lorentzian for all , unifying discrete convexity with Lorentzian polynomial theory and resolving a longstanding question of Eur–Huh. A key analytic tool is a rank-1 bound for the leaf-distance matrix of ultrametric trees, refining Graham–Pollak’s single-positive-eigenvalue result and enabling an inductive Lorentzian argument. As consequences, the authors establish strengthened polynomial inequalities for Mason-type log-concavity in both valuated and ordinary matroids, answering open questions by Giansiracusa–Rincón–Schleis–Ulirsch and Pak. Overall, the work builds a bridge between matroid theory, discrete convex analysis, and Lorentzian polynomials, yielding new inequalities and a framework with potential impacts on combinatorial Hodge theory and matroid valuation.

Abstract

We show that a set function satisfies the gross substitutes property if and only if its homogeneous generating polynomial is a Lorentzian polynomial for all positive , answering a question of Eur-Huh. We achieve this by giving a rank 1 upper bound for the distance matrix of an ultrametric tree, refining a classical result of Graham-Pollak. This characterization enables us to resolve two open problems that strengthen Mason's log-concavity conjectures for the number of independent sets of a matroid: one posed by Giansiracusa-Rincón-Schleis-Ulirsch for valuated matroids, and another posed by Pak for ordinary matroids.
Paper Structure (11 sections, 10 theorems, 60 equations, 1 figure)

This paper contains 11 sections, 10 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Inequalities for $\text{M}^\natural$-concave set functions
  3. Polynomial inequalities for matroids
  4. An inequality for ultrametric trees
  5. Trees
  6. Ultrametric tree matrices
  7. Equality cases
  8. Lorentzian polynomials from trees
  9. Lorentzian polynomials and discrete convex analysis
  10. Lorentzian property of the homogeneous generating polynomial $Z_{q,\nu}$
  11. Polynomial log concavity

Key Result

Theorem 1.4

For any $\text{M}^\natural$-concave function $\nu\colon 2^{E} \to \mathbb{R} \cup \{-\infty\}$, we have

Figures (1)

  • Figure 1: Non-binary trees can be seen as binary by adding edges of length $0$.

Theorems & Definitions (22)

  • Example 1.1: Matroid independent sets
  • Example 1.2: Matroid rank functions
  • Example 1.3: Valuated matroid independent sets
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Definition 2.1
  • Proposition 2.2
  • ...and 12 more