The Bhargava Greedoid as a Gaussian Elimination Greedoid

Abstract

Inspired by Manjul Bhargava's theory of generalized factorials, Grinberg and Petrov have defined the Bhargava greedoid - a greedoid (a matroid-like set system on a finite set) assigned to any "ultra triple" (a somewhat extended variant of a finite ultrametric space). Here we show that the Bhargava greedoid of a finite ultra triple is always a Gaussian elimination greedoid over any sufficiently large (e.g., infinite) field; this is a greedoid analogue of a representable matroid. We find necessary and sufficient conditions on the size of the field to ensure this.