Bouquet algebra of toric ideals

To any toric ideal IA, encoded by an integer matrix A, we associate a matroid structure called the bouquet graph of A and introduce another toric ideal called the bouquet ideal of A. We show how these objects capture the essential combinatorial and algebraic information about IA. Passing from the toric ideal to its bouquet ideal via the graph theoretic properties of the bouquet graph allows us to classify several cases. For example, on the one end of the spectrum, there are ideals that we call 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, Gröbner, 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 Gröbner basis, any reduced Gröbner basis and any minimal generating set coincide.