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Matroids in OSCAR

Daniel Corey, Lukas Kühne, Benjamin Schröter

TL;DR

This work presents OSCAR's matroid module, focusing on efficiently computing realization spaces $\mathcal{R}(\mathsf{M};\mathbb{F})$ and Chow rings $\mathrm{A}(\mathsf{M})$ for matroids. It introduces an affine-coordinate-ring algorithm that encodes realizations via a determinant matrix $X$, imposes basis-based polynomial constraints, factors out scalings, and uses Gröbner-basis reductions to decide realizability through the ring $S_{\mathsf{M}}$, with speedups such as basis selection and ambient-space reduction. A broad set of examples (Fano, non-Fano, Vámos, Möbius–Kantor, K4, Pappus) demonstrates field-dependent realizability, single-point realizations, and geometric realization spaces, while Mnëv universality and polyDB integration illustrate the depth and breadth of realizability phenomena. The Chow-ring treatment defines $\mathrm{A}(\mathsf{M})$ and its Lefschetz structure, enabling computation of invariants and establishing connections between matroid invariants (via the Tutte and characteristic polynomials) and geometric properties, including the Happel–Adiprasito–Huh–Katz results on log-concavity through the Hodge–Riemann framework. Overall, the paper provides concrete computational tools that bridge combinatorics and algebraic geometry, enabling systematic study of realizability and Chow-theoretic properties with practical software support.

Abstract

OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we present parts of the module handeling matroids in OSCAR, which will appear as a chapter of the upcoming OSCAR book. A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear dependency in vector spaces as well as many aspects of graph theory. Moreover, matroids form a cornerstone of tropical geometry and a deep link between algebraic geometry and combinatorics. Our focus lies in particular on computing the realization space and the Chow ring of a matroid.

Matroids in OSCAR

TL;DR

This work presents OSCAR's matroid module, focusing on efficiently computing realization spaces and Chow rings for matroids. It introduces an affine-coordinate-ring algorithm that encodes realizations via a determinant matrix , imposes basis-based polynomial constraints, factors out scalings, and uses Gröbner-basis reductions to decide realizability through the ring , with speedups such as basis selection and ambient-space reduction. A broad set of examples (Fano, non-Fano, Vámos, Möbius–Kantor, K4, Pappus) demonstrates field-dependent realizability, single-point realizations, and geometric realization spaces, while Mnëv universality and polyDB integration illustrate the depth and breadth of realizability phenomena. The Chow-ring treatment defines and its Lefschetz structure, enabling computation of invariants and establishing connections between matroid invariants (via the Tutte and characteristic polynomials) and geometric properties, including the Happel–Adiprasito–Huh–Katz results on log-concavity through the Hodge–Riemann framework. Overall, the paper provides concrete computational tools that bridge combinatorics and algebraic geometry, enabling systematic study of realizability and Chow-theoretic properties with practical software support.

Abstract

OSCAR is an innovative new computer algebra system which combines and extends the power of its four cornerstone systems - GAP (group theory), Singular (algebra and algebraic geometry), Polymake (polyhedral geometry), and Antic (number theory). Here, we present parts of the module handeling matroids in OSCAR, which will appear as a chapter of the upcoming OSCAR book. A matroid is a fundamental and actively studied object in combinatorics. Matroids generalize linear dependency in vector spaces as well as many aspects of graph theory. Moreover, matroids form a cornerstone of tropical geometry and a deep link between algebraic geometry and combinatorics. Our focus lies in particular on computing the realization space and the Chow ring of a matroid.
Paper Structure (6 sections, 1 theorem, 19 equations, 1 figure)

This paper contains 6 sections, 1 theorem, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basics
  3. Realizability
  4. Algorithmic Framework
  5. Examples
  6. Chow rings of Geometries

Key Result

Theorem 4.1

Let $\ell\in\mathrm{A}^1(\mathsf{M})$ be a Lefschetz element. Then the following three properties hold. For every non-negative integer $k\leq \tfrac{\mathrm{rk}(E)-1}{2}$, the bilinear pairing is non-degenerate. For every non-negative integer $k\leq \tfrac{\mathrm{rk}(E)-1}{2}$, the multiplication map is an isomorphism. For every non-negative integer $k\leq \tfrac{\mathrm{rk}(E)-1}{2}$, the bili

Figures (1)

  • Figure 1: Projective realizations of the Fano, non-Fano and Möbius--Kantor matroids. Points connected by an arc are interpreted as colinear.

Theorems & Definitions (9)

  • Definition 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Theorem 4.1: AHK