Bouquet algebra of toric ideals

TL;DR

This paper introduces the bouquet algebra framework for toric ideals, using matroid structures to classify and analyze their algebraic and combinatorial properties, enabling new insights and constructions.

Contribution

It defines the bouquet graph and bouquet ideal, providing a novel approach to classify and construct toric ideals with specific properties.

Findings

Bouquet graph captures essential combinatorial information.

Classification of stable and special toric ideals.

Bouquet framework characterizes ideals with coinciding bases.

Abstract

To any toric ideal $I_A$, encoded by an integer matrix $A$, we associate a matroid structure called {\em the bouquet graph} of $A$ and introduce another toric ideal called {\em the bouquet ideal} of $A$. We show how these objects capture the essential combinatorial and algebraic information about $I_A$. Passing from the toric ideal to its bouquet ideal reveals a structure that allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call {\em stable}, for which bouquets capture the complexity of various generating sets as well as the minimal free resolution. On the other end of the spectrum lie toric ideals whose various bases (e.g., minimal generating sets, Gr\"obner, Graver bases) coincide. Apart from allowing for classification-type results, bouquets provide a new way to construct families of examples of toric ideals with various interesting properties, such as robustness, genericity, and unimodularity. The new bouquet framework can be used to provide a characterization of toric ideals whose Graver basis, the universal Gr\"obner basis, any reduced Gr\"obner basis and any minimal generating set coincide.