Hypergraph encodings of arbitrary toric ideals

TL;DR

This paper explores how hypergraph-based encodings can represent any toric ideal, revealing their universality and introducing a polarization operation that preserves key properties.

Contribution

It demonstrates that hypergraph encodings can represent all toric ideals of general matrices and introduces a polarization operation maintaining combinatorial and homological features.

Findings

Hypergraph encodings can represent any toric ideal from a general matrix.

Unbounded degrees of generating sets are shown for various bases.

A polarization operation preserves combinatorial and homological properties.

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 $0/1$ 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.