The Bhargava greedoid as a Gaussian elimination greedoid
arXiv:2001.05535 · doi:10.37236/11222
Abstract
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.
71 pages. Pages 1-39 are the core; the rest proves the optimality of the bound (under certain conditions) and fills in some very basic details. See http://www.cip.ifi.lmu.de/~grinberg/algebra/fps20gfv.pdf for a streamlined survey. Comments are welcome! v5 corrects a few typos and adds Example 2.5. This is NOT the published version!