The Bhargava greedoid as a Gaussian elimination greedoid

01/15/2020

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by   Darij Grinberg, et al.

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Inspired by Manjul Bhargava's theory of generalized factorials, Fedor Petrov and the author 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.