On the complex-representable excluded minors for real-representability

TL;DR

The paper demonstrates that every real-representable matroid can be found as a minor within a complex-representable excluded minor for real-representability, extending to certain field extensions.

Contribution

It establishes a general framework linking representability over different fields and shows how matroids relate as minors within excluded minors across fields.

Findings

Every real-representable matroid is a minor of a complex-representable excluded minor.

The result extends to infinite fields and their extensions where representability differs.

Provides a new perspective on the structure of excluded minors across fields.

Abstract

We show that each real-representable matroid is a minor of a complex-representable excluded minor for real-representability. More generally, for an infinite field $\mathbb{F}_1$ and a field extension $\mathbb{F}_2$, if $\mathbb{F}_1$-representability is not equivalent to $\mathbb{F}_2$-representability, then each $\mathbb{F}_1$-representable matroid is a minor of a $\mathbb{F}_2$-representable excluded minor for $\mathbb{F}_1$-representability.