Effective versions of two theorems of Rado

TL;DR

This paper provides bounds on the minimal positive characteristic and smallest field size needed for a representable matroid to be represented over a field, advancing understanding of matroid representability conditions.

Contribution

It introduces effective bounds on the characteristic and field size for representability of matroids, refining previous theoretical results.

Findings

Bounds on minimal positive characteristic for matroid representability

Bounds on smallest field size for matroid representability

Improved understanding of conditions for matroid representation

Abstract

Let $M$ be a representable matroid on $n$ elements. We give bounds, in terms of $n$, on the least positive characteristic and smallest field over which $M$ is representable.