On robustness and related properties on toric ideals

TL;DR

This paper explores the properties of robust and generalized robust toric ideals, providing combinatorial characterizations for graph-based ideals and algebraic characterizations for numerical semigroup-based ideals, revealing new classes and properties.

Contribution

It offers the first comprehensive analysis of robustness properties in toric ideals from graphs and numerical semigroups, including characterizations and new examples.

Findings

Characterization of graphs producing robust and generalized robust quadratic toric ideals

Identification of free numerical semigroups via initial ideals as complete intersections

Proving that generalized robustness corresponds to having a unique Betti element

Abstract

A toric ideal is called robust if its universal Gr\"obner basis is a minimal set of generators, and is called generalized robust if its universal Gr\"obner basis equals its universal Markov basis (the union of all its minimal sets of binomial generators). Robust and generalized robust toric ideals are both interesting from both a Commutative Algebra and an Algebraic Statistics perspective. However, only a few nontrivial examples of such ideals are known. In this work we study these properties for toric ideals of both graphs and numerical semigroups. For toric ideals of graphs, we characterize combinatorially the graphs giving rise to robust and to generalized robust toric ideals generated by quadratic binomials. As a byproduct, we obtain families of Koszul rings. For toric ideals of numerical semigroups, we determine that one of its initial ideals is a complete intersection if and only if the semigroup belongs to the so-called family of free numerical semigroups. Hence, we characterize all complete intersection numerical semigroups which are minimally generated by one of its Gr\"obner basis and, as a consequence, all the Betti numbers of the toric ideal and its corresponding initial ideal coincide. Moreover, also for numerical semigroups, we prove that the ideal is generalized robust if and only if the semigroup has a unique Betti element and that there are only trivial examples of robust ideals. We finish the paper with some open questions.