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 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 Gröbner 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.2010 Mathematics Subject Classification. 14M25, 13P10, 05C65, 13D02.

Koszul blowup algebras associated to three-dimensional Ferrers diagrams

“…As a result, we consider the three-dimensional Ferrers diagram and the monomial ideal associated to it. Notice that the toric ideals of toric rings generated by squarefree monomials of degree ≤ 3 are as complicated as any arbitrary toric ideals by [25,Theorem 3.2].…”

Algebraic statistics, tables, and networks: The Fienberg advantage

“…Indeed, the results of [PRF10] provided an applied motivation for the systematic study of toric ideals of hypergraphs in the field of combinatorial commutative algebra (see e.g. [GP13], [HT08], [PS14], [PTV19]).…”

Bouquet algebra of toric ideals

“…This result has nice consequences in integer programming, implying two universality results [31,Corollaries 3.3,3.4] about the unboundedness of the degrees of all elements in a minimal Markov basis (and similarly about Graver basis, universal Gröbner basis, indispensable binomials respectively). Finally, the most important consequence of our results is a kind of polarization-type operation, see [31,Theorem 4.2], which allows us, in particular, to answer problems like [10, Question 6.1], for arbitrary toric ideals, by reducing them to toric ideals of hypergraphs. Details of those results are beyond the scope of the present manuscript.…”

“…This new toric ideal encodes the basic properties of I A via a structural decomposition of A. The terminology we use comes from the applications to hypergraphs (see [31,Section 2]). The graph's connected components alone were used in [12], under the name coparallelism classes, to provide combinatorial descriptions of self-dual projective toric varieties associated to a non-pyramidal configuration, see [12,Theorem 4.16].…”