The Bhargava greedoid as a Gaussian elimination greedoid
Darij Grinberg
TL;DR
This work establishes that the Bhargava greedoid attached to a finite $\mathbb{V}$-ultra triple is always representable as a Gaussian elimination greedoid over any sufficiently large field. The authors introduce valadic $\mathbb{V}$-ultra triples to prove a determinantal representation for the Bhargava greedoid in the valadic setting, then show any finite $\mathbb{V}$-ultra triple is isomorphic to a valadic one over a large field, yielding representability. A stronger bound using the maximum clique size $\mathrm{mcs}(E,w,d)$ is proved, and the converse is established in the constant-weight case. The results bridge Bhargava’s generalized factorials with greedoid/linear-algebraic representability, with potential applications in phylogenetics and combinatorial optimization. The paper also develops a substantial toolkit—open/closed balls, clique decompositions, determinant criteria, and determinant-based Plücker identities—to underpin the representation theorems and the converse bounds.
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.