The virtual Euler characteristic for binary matroids

TL;DR

This paper introduces a virtual Euler characteristic for finite sets of matroids, providing a simple formula for binary matroids and relating it to Grassmannian point counts over finite fields.

Contribution

It defines a new virtual Euler characteristic for matroids and derives a formula specifically for binary matroids, connecting it to algebraic geometry and finite field point counts.

Findings

Derived a formula for the virtual Euler characteristic of binary matroids.

Connected the characteristic to point counts on Grassmannians over finite fields.

Suggested future research directions in matroid theory and algebraic geometry.

Abstract

Inspired by Kontsevich's graphic orbifold Euler characteristic we define a virtual Euler characteristic for any finite set of isomorphism classes of matroids of rank $r$. Our main result provides a simple formula for the virtual Euler characteristic for the set of isomorphism classes of matroids of rank $r$ realizable over $\mathbb{F}_2$ (i.e., binary matroids). We prove this formula by relating the virtual Euler characteristic for binary matroids to the point counts of certain subsets of Grassmanians over finite fields. We conclude by providing several follow-up questions in relation to matroids realizable over other finite prime fields, matroid homology, and beta invariants.