Certifying Non-representability of Matroids Over Prime Fields

TL;DR

This paper demonstrates that verifying a matroid's non-representability over a prime field can be achieved efficiently with a quadratic number of rank evaluations, simplifying the certification process.

Contribution

It introduces an $O(n^2)$ rank evaluation method to certify non-representability of matroids over prime fields, improving efficiency over previous approaches.

Findings

Non-representability can be certified with quadratic rank evaluations.

The method applies to matroids with $n$ elements over prime fields.

Efficiency in certification process is significantly improved.

Abstract

It is proved that, for a prime number $p$, showing that an $n$-element matroid is not representable over $GF(p)$ requires only $O(n^2)$ rank evaluations.