Hypergraph encodings of arbitrary toric ideals
Abstract
Relying on the combinatorial classification of toric ideals using their bouquet structure, we focus on toric ideals of hypergraphs and study how they relate to general toric ideals. We show that hypergraphs exhibit a surprisingly general behavior: the toric ideal associated to any general matrix can be encoded by that of a matrix, while preserving the essential combinatorics of the original ideal. We provide two universality results about the unboundedness of degrees of various generating sets: minimal, Graver, universal Gr\"obner bases, and indispensable binomials. Finally, we provide a polarization-type operation for arbitrary positively graded toric ideals, which preserves all the combinatorial signatures and the homological properties of the original toric ideal.
Cite
@article{arxiv.1711.04354,
title = {Hypergraph encodings of arbitrary toric ideals},
author = {Sonja Petrović and Apostolos Thoma and Marius Vladoiu},
journal= {arXiv preprint arXiv:1711.04354},
year = {2017}
}
Comments
30 pages, 7 figures; this paper is an extended version of the last 3 sections of arXiv:1507.02740v2. We have added 2 new results Corollaries 3.3,3.4 and a new section