Related papers: Generalized robust toric ideals
Let I be a toric ideal. We say I is robust if its universal Groebner basis is a minimal generating set. We show that any robust toric ideal arising from a graph G is also minimally generated by its Graver basis. We then completely…
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…
A homogeneous ideal is robust if its universal Gr\"obner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable…
The universal Gr\"{o}bner basis of $I$, is a Gr\"{o}bner basis for $I$ with respect to all term orders simultaneously. Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the elements of the universal…
Describing families of ideals that are minimally generated by at least one, or by all, of their reduced Gr\"obner bases is a central topic in commutative algebra. In this paper, we address this problem in the context of toric ideals of…
In this work, we study the equivalence of various robustness properties of toric ideals of weighted oriented graphs. For any weighted oriented graph $D$, if its toric ideal $I_D$ is generalized robust (or weakly robust), then we show that…
We call an ideal in a polynomial ring robust if it can be minimally generated by a universal Gr\"obner basis. In this paper we show that robust toric ideals generated by quadrics are essentially determinantal. We then discuss two possible…
To every toric ideal one can associate an oriented matroid structure, consisting of a graph and another toric ideal, called bouquet ideal. The connected components of this graph are called bouquets. Bouquets are of three types; free, mixed…
We describe explicitly all multisets of weights whose defining projective toric varieties are self-dual. In addition, we describe a remarkable and unexpected combinatorial behaviour of the defining ideals of these varieties. The toric ideal…
The universal Gr\"obner basis of an ideal is a Gr\"obner basis with respect to all term orders simultaneously. The aim of this paper is to present an algorithmic approach to compute the universal Gr\"obner basis for the toric ideal…
Strongly robust toric ideals are the toric ideals for which the set of indispensable binomials is the Graver basis. The strongly robust simplicial complex $\Delta _T$ of a simple toric ideal $I_T$ determines the strongly robust property for…
In this article, we investigate the strongly robust property of toric ideals associated with weighted oriented graphs. We establish that the toric ideals of a broad class of monomial ideals are strongly robust; this class encompasses the…
In this paper, we discuss the normality of the toric rings of stable set polytopes, and the set of generators and Gr\"obner bases of toric ideals of stable set polytopes by using the results on that of edge polytopes of finite nonsimple…
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…
A numerical semigroup is said to be universally free if it is free for any possible arrangement of its minimal generating set. In this work, we establish that toric ideals associated with universally free numerical semigroups can be…
Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gr\"obner basis and the Graver basis. While the former is generally a proper…
A universal Gr\"obner basis of an ideal is the union of all its reduced Gr\"obner bases. It is contained in the Graver basis, the set of all primitive elements. Obtaining an explicit description of either of these sets, or even a sharp…
Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which…
Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$.
We investigate the computational complexity of problems on toric ideals such as normal forms, Gr\"obner bases, and Graver bases. We show that all these problems are strongly NP-hard in the general case. Nonetheless, we can derive efficient…