Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rank functions and invariants of delta-matroids

Matt Larson

TL;DR

This paper develops a rank-function framework for delta-matroids by axiomatizing $g_D$ and defines the $U$-polynomial $U_D(u,v)$ as a Tutte-like invariant encoding independent-set structure. It analyzes how rank functions transform under delta-matroid operations (deletion, contraction, projection, product, twisting) and introduces an alternative normalization $h_D$, linking to enveloping matroids. The independence complex and an activity-based expansion of the $U$-polynomial are established, with enveloping matroids providing a bridge to Lorentzian polynomial techniques. Using Lorentzian methods, the paper proves log-concavity-type inequalities for coefficients of $U_D(u,0)$ in cases with enveloping matroids, highlighting deep connections between delta-matroid theory, algebraic geometry, and combinatorial invariants.

Abstract

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid, and we prove a log-concavity result for that evaluation using the theory of Lorentzian polynomials.

Rank functions and invariants of delta-matroids

TL;DR

This paper develops a rank-function framework for delta-matroids by axiomatizing and defines the -polynomial as a Tutte-like invariant encoding independent-set structure. It analyzes how rank functions transform under delta-matroid operations (deletion, contraction, projection, product, twisting) and introduces an alternative normalization , linking to enveloping matroids. The independence complex and an activity-based expansion of the -polynomial are established, with enveloping matroids providing a bridge to Lorentzian polynomial techniques. Using Lorentzian methods, the paper proves log-concavity-type inequalities for coefficients of in cases with enveloping matroids, highlighting deep connections between delta-matroid theory, algebraic geometry, and combinatorial invariants.

Abstract

In this note, we give a rank function axiomatization for delta-matroids and study the corresponding rank generating function. We relate an evaluation of the rank generating function to the number of independent sets of the delta-matroid, and we prove a log-concavity result for that evaluation using the theory of Lorentzian polynomials.
Paper Structure (9 sections, 21 theorems, 31 equations)

This paper contains 9 sections, 21 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Rank functions of delta-matroids
  3. Compatibility with delta-matroid operations
  4. An alternative normalization
  5. The $U$-polynomial
  6. The independence complex of a delta-matroid
  7. The activity expansion of the $U$-polynomial
  8. Enveloping matroids
  9. Lorentzian polynomials

Key Result

Theorem 1.2

A function $g \colon \operatorname{AdS}_n \to \mathbb{Z}$ is the rank function of a delta-matroid if and only if Furthermore, $D$ is even if and only if

Theorems & Definitions (48)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Example 1.4
  • Conjecture 1.5
  • Theorem 1.6
  • proof : Proof of Theorem \ref{['thm:rank']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • ...and 38 more