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Linearizing Algebraic Matroids

Zvi Rosen, Jessica Sidman, Louis Theran

Abstract

Although algebraic matroids were discovered in the 1930s, interest in them was largely dormant until their recent use in applications of algebraic geometry. Because nonlinear algebra is computationally challenging, it is easier to work with an isomorphic linear matroid if one exists. We describe an explicit construction that produces a linear representation over an algebraically closed field of characteristic zero starting with the data used in applications. We will also discuss classical examples of algebraic matroids in the modern language of polynomial ideals, illustrating how the existence of an isomorphic linear matroid depends on properties of the field.

Linearizing Algebraic Matroids

Abstract

Although algebraic matroids were discovered in the 1930s, interest in them was largely dormant until their recent use in applications of algebraic geometry. Because nonlinear algebra is computationally challenging, it is easier to work with an isomorphic linear matroid if one exists. We describe an explicit construction that produces a linear representation over an algebraically closed field of characteristic zero starting with the data used in applications. We will also discuss classical examples of algebraic matroids in the modern language of polynomial ideals, illustrating how the existence of an isomorphic linear matroid depends on properties of the field.

Paper Structure

This paper contains 17 sections, 6 theorems, 32 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Best Case: Algebraically Closed and Characteristic Zero
  3. Algebraic varieties
  4. Differentials
  5. Specialization
  6. Proof of Theorem \ref{['thm: K/k-linear']}
  7. (1)
  8. (2)
  9. What if the Field is Not Algebraically Closed?
  10. The Wild World of Characteristic $p$
  11. The Non-Pappus Matroid is algebraic but not linear
  12. Frobenius Flocks
  13. Acknowledgments and declarations
  14. Acknowledgments
  15. Funding
  16. ...and 2 more sections

Key Result

Theorem 1.10

Let $k$ be a field of characteristic zero and $M$ be a $k$-algebraic matroid.

Figures (1)

  • Figure 1: Pappus's Theorem and the non-Pappus matroid

Theorems & Definitions (23)

  • Example 1.1: cf owl.pdf
  • Definition 1.2
  • Example 1.3
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • Example 1.7
  • Theorem 1.10
  • Lemma 2.1
  • Lemma 2.2
  • ...and 13 more