Linearizing Algebraic Matroids

TL;DR

This paper presents a method to explicitly construct linear representations of algebraic matroids over algebraically closed fields of characteristic zero, facilitating their use in algebraic geometry applications.

Contribution

It introduces an explicit construction for linearizing algebraic matroids from application data, bridging nonlinear algebra challenges with linear algebra techniques.

Findings

Construction works over algebraically closed fields of characteristic zero

Existence of linear representations depends on properties of the field

Classical algebraic matroids can be represented linearly using polynomial ideals

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.