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Tropical Geometry of Rado Matroids

Calum Buchanan, Richard Danner

Abstract

In this note, we characterize the products of simplicial generators for the Chow ring of a loopless matroid, extending a result of Backman, Eur, and Simpson. We prove that the stable intersection of a collection of tropical hyperplanes centered at the origin with the Bergman fan of a matroid is the Bergman fan of the dual of a certain Rado matroid.

Tropical Geometry of Rado Matroids

Abstract

In this note, we characterize the products of simplicial generators for the Chow ring of a loopless matroid, extending a result of Backman, Eur, and Simpson. We prove that the stable intersection of a collection of tropical hyperplanes centered at the origin with the Bergman fan of a matroid is the Bergman fan of the dual of a certain Rado matroid.
Paper Structure (1 section, 5 theorems, 5 equations, 1 figure)

This paper contains 1 section, 5 theorems, 5 equations, 1 figure.

Table of Contents

  1. Acknowledgement

Key Result

Theorem 1

Let $M$ be an arbitrary matroid on a set $Y$, and let $\mathcal{X}$ be a collection of subsets $X_1, \ldots, X_m$ of $Y$. There exists a transversal of $\mathcal{X}$ which is independent in $M$ if and only if $\mathop{\mathrm{rk}}\nolimits_M(\cup_{j\in J} X_j) \geq |J|$ for all $J \subseteq \{1, \ld

Figures (1)

  • Figure 1: The graph $G(\mathcal{A})$, where $E = \{1, \ldots, 7\}$, $\mathcal{A} = \{A_1, A_2\}$, $A_1 = \{2,3,4\}$, and $A_2 = \{4,6\}$.

Theorems & Definitions (10)

  • Theorem 1: Rado's theorem rado1942theorem
  • Definition 1
  • Definition 2
  • Theorem 2
  • Corollary 3
  • proof : Proof of Theorem \ref{['thm:coradointersections']}
  • Example 1
  • Corollary 4: fink2022presentations
  • Corollary 5: Dragon-Hall-Rado theorem backman2023simplicial
  • proof