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Paving Matroids: Defining Equations and Associated Varieties

Emiliano Liwski, Fatemeh Mohammadi

Abstract

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of these varieties across any dimension. Additionally, we explain the relationship between polynomials constructed using these different methods. We then compute a comprehensive and finite set of defining equations for matroid varieties associated with specific classes of paving matroids. Finally, we focus on the class of paving matroids of rank $3$, known as point-line configurations, which essentially contain simple matroids of rank $3$. Furthermore, we provide a decomposition for the associated circuit variety of point-line configurations, where all points have a degree less than $3$. Lastly, we present several examples applying our results and compare them with the known cases in the literature.

Paving Matroids: Defining Equations and Associated Varieties

Abstract

We study paving matroids, their realization spaces, and their closures, along with matroid varieties and circuit varieties. Within this context, we introduce three distinct methods for generating polynomials within the associated ideals of these varieties across any dimension. Additionally, we explain the relationship between polynomials constructed using these different methods. We then compute a comprehensive and finite set of defining equations for matroid varieties associated with specific classes of paving matroids. Finally, we focus on the class of paving matroids of rank , known as point-line configurations, which essentially contain simple matroids of rank . Furthermore, we provide a decomposition for the associated circuit variety of point-line configurations, where all points have a degree less than . Lastly, we present several examples applying our results and compare them with the known cases in the literature.
Paper Structure (15 sections, 42 theorems, 95 equations, 2 figures)

This paper contains 15 sections, 42 theorems, 95 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Realization spaces of matroids and associated varieties
  4. Grassmann-Cayley algebra
  5. Computing lifting polynomials
  6. Lifting ideals of paving matroids
  7. Infinitesimal motion
  8. Liftability property
  9. Lifting polynomials
  10. Decomposing circuit varieties
  11. Computing graph polynomials
  12. Polynomials arising from graphs
  13. Polynomials arising from circuits
  14. Relation between lifting polynomials and graph polynomials
  15. Examples and applications

Key Result

Theorem 1.2

Let $M$ be an $n$-paving matroid and $N$ be a submatroid of $M$ with full rank. Then, for any $q\in \mathbb{C}^{n}$, the $|N|-n+1$ minors of the matrix $\mathcal{M}_{q}(N)$ are polynomials within $I_{M}$.

Figures (2)

  • Figure 1: (Left) Three concurrent lines; (Center) Quadrilateral set; (Right) Pascal configuration.
  • Figure 2: (Left) $3\times 4$ grid; (Center) Matroid in Example \ref{['Ex:8 nodes']}; (Right) Matroid in Example \ref{['Ex:7 nodes']}.

Theorems & Definitions (123)

  • Theorem 1.2: Theorem \ref{['sub']}
  • Theorem 1.3: Theorem \ref{['theo lift']}
  • Theorem 1.4
  • Theorem 1.5: Theorem \ref{['inc']} and Corollary \ref{['fin']}
  • Definition 2.1: Matroid variety
  • Definition 2.2: Circuit and basis ideals
  • Proposition 2.3
  • Definition 2.4: Paving matroid
  • Lemma 2.5
  • proof
  • ...and 113 more