Toric Ideals Research Articles

Minimal generators of toric ideals of graphs

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 .

Toric ideals of lattice path matroids and polymatroids

White has conjectured that the toric ideal of a matroid is generated by quadric binomials corresponding to symmetric basis exchanges. We prove a stronger version of this conjecture for lattice path polymatroids by constructing a monomial order under which these sets of quadrics form Gröbner bases. We then introduce a larger class of polymatroids for which an analogous theorem holds. Finally, we obtain the same result for lattice path matroids as a corollary.

On the universal Gröbner bases of toric ideals of graphs

The universal Gröbner basis of an ideal is a Gröbner basis with respect to all term orders simultaneously. We characterize in graph theoretical terms the elements of the universal Gröbner basis of the toric ideal of a graph. We also provide a new degree bound. Finally, we give examples of graphs for which the true degrees of their circuits are less than the degrees of some elements of the Graver basis.

Toric ideals for high Veronese subrings of toric algebras

We prove that the defining ideal of a sufficiently high Veronese subring of a toric algebra admits a quadratic Gröbner basis consisting of binomials. More generally, we prove that the defining ideal of a sufficiently high Veronese subring of a standard graded ring admits a quadratic Gröbner basis. We give a lower bound on $d$ such that the defining ideal of $d$th Veronese subring admits a quadratic Gröbner basis. Eisenbud–Reeves–Totaro stated the same theorem without a proof with some lower bound on $d$. In many cases, our lower bound is less than Eisenbud–Reeves–Totaro’s lower bound.

Simplicial ideals, 2-linear ideals and arithmetical rank

In the first part of this paper we study scrollers and linearly joined varieties. Scrollers were introduced in Barile and Morales (2004) [BM4], linearly joined varieties are an extension of scrollers and were defined in Eisenbud et al. (2005) [EGHP], there they proved that scrollers are defined by homogeneous ideals having a 2-linear resolution. A particular class of varieties, of important interest in classical Geometry are Cohen–Macaulay varieties of minimal degree, they were classified geometrically by the successive contribution of Del Pezzo (1885) [DP], Bertini (1907) [B], and Xambo (1981) [X] and algebraically in Barile and Morales (2000) [BM2]. They appear naturally studying the fiber cone of a codimension two toric ideals Morales (1995) [M], Gimenez et al. (1993, 1999) [GMS1,GMS2], Barile and Morales (1998) [BM1], Ha (2006) [H], Ha and Morales (2009) [HM]. Let S be a polynomial ring and I ⊂ S a homogeneous ideal defining a sequence of linearly joined varieties. • We compute depth S / I . • We prove that c ( V ) = depth S / I − 1 , where c ( V ) is the connectedness dimension of the algebraic set defined by I . • We characterize sets of generators of I , and give an effective algorithm to find equations, as an application we prove that ara ( I ) = projdim ( S / I ) in the case where V is a union of linear spaces, in particular this applies to any square free monomial ideal having a 2-linear resolution. • In the case where V is a union of linear spaces, the ideal I can be characterized by a tableau, which is an extension of a Ferrer (or Young) tableau. All these results are new, and extend results in Barile and Morales (2004) [BM4], Eisenbud et al. (2005) [EGHP].

Arithmetical rank of toric ideals associated to graphs

Let I G C K[x 1 , ... , x m ] be the toric ideal associated to a finite graph G. In this paper we study the binomial arithmetical rank and the G-homogeneous arithmetical rank of I G in 2 cases: (1) G is bipartite, (2) I G is generated by quadratic binomials. In both cases we prove that the binomial arithmetical rank and the G-homogeneous arithmetical rank coincide with the minimal number of generators of I G .

GENERALIZED REDUCTION TO COMPUTE TORIC IDEALS

Toric ideals have many applications including solving integer programs. Several algorithms for computing the toric ideal of an integer matrix are available in the literature. Since it is an NP hard problem the present approaches can only solve relatively small problems. We propose a new approach which improves upon a well known technique to compute ideal saturation, which leads to a faster algorithm to compute toric ideals.

The Toric Ideal of a Matroid of Rank 3 is Generated by Quadrics

White conjectured that the toric ideal associated with the basis of a matroid is generated by quadrics corresponding to symmetric exchanges. We present a combinatorial proof of White's conjecture for matroids of rank 3 by using a lemma proposed by Blasiak.

Toric Ideals of Flow Polytopes

We show that toric ideals of flow polytopes are generated in degree $3$. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope $B_n$ have at most degree $n$. We show that this bound is sharp for some revlex term orders. For $(m \times n)$-transportation polytopes, a similar result holds: they have Gröbner bases of at most degree $\lfloor mn/2 \rfloor$. We construct a family of examples, where this bound is sharp. Nous démontrons que les idéaux toriques des polytopes de flot sont engendrés par un ensemble de degré $3$. Cela a été conjecturé par Diaconis et Eriksson pour le cas particulier du polytope de Birkhoff. Notre preuve utilise une méthode de subdivision par hyperplans, développée par Haase et Paffenholz. Il est bien connu que les bases de Gröbner revlex réduite du polytope de Birkhoff $B_n$ sont au plus de degré $n$. Nous démontrons que cette borne est optimale pour quelques ordres revlex. Pour les polytopes de transportation de dimension $(m \times n)$, il existe un résultat similaire : leurs bases de Gröbner sont au plus de degré $\lfloor mn/2 \rfloor$. Nous construisons une famille d'exemples pour lesquels cette borne est atteinte.

Additive edge labelings

Let G = ( V , E ) be a graph and d a positive integer. We study the following problem: for which labelings f E : E → Z d is there a labeling f V : V → Z d such that f E ( i , j ) = f V ( i ) + f V ( j ) ( mod d ) , for every edge ( i , j ) ∈ E ? We also explore the connections of the equivalent multiplicative version to toric ideals. We derive a polynomial algorithm to answer these questions and to obtain all possible solutions.

Perturbation method for determining the group of invariance of hierarchical models

We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in [S. Aoki, A. Takemura, The largest group of invariance for Markov bases and toric ideals, J. Symbolic Comput. 43 (5) (2008) 342–358]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof by Aoki and Takemura which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.

Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

Let $\mathcal{C}$ be a uniform clutter and let A be the incidence matrix of $\mathcal{C}$ . We denote the column vectors of A by v 1,…,v q . Under certain conditions we prove that $\mathcal{C}$ is vertex critical. If $\mathcal{C}$ satisfies the max-flow min-cut property, we prove that A diagonalizes over ℤ to an identity matrix and that v 1,…,v q form a Hilbert basis. We also prove that if $\mathcal{C}$ has a perfect matching such that $\mathcal{C}$ has the packing property and its vertex covering number is equal to 2, then A diagonalizes over ℤ to an identity matrix. If A is a balanced matrix we prove that any regular triangulation of the cone generated by v 1,…,v q is unimodular. Some examples are presented to show that our results only hold for uniform clutters. These results are closely related to certain algebraic properties, such as the normality or torsion-freeness, of blowup algebras of edge ideals and to finitely generated abelian groups. They are also related to the theory of Gröbner bases of toric ideals and to Ehrhart rings.

Ring graphs and complete intersection toric ideals

We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation.

Quadratic Gröbner bases for smooth 3×3 transportation polytopes

The toric ideals of 3×3 transportation polytopes $\mathsf{T}_{\mathbf{rc}}$ are quadratically generated. The only exception is the Birkhoff polytope B 3. If $\mathsf{T}_{\mathbf{rc}}$ is not a multiple of B 3, these ideals even have square-free quadratic initial ideals. This class contains all smooth 3×3 transportation polytopes.

Unmixed bipartite graphs and sublattices of the Boolean lattices

The correspondence between unmixed bipartite graphs and sublattices of the oolean lattice is discussed. By using this correspondence, we show the existence of squarefree quadratic initial ideals of toric ideals arising from minimal vertex covers of unmixed bipartite graphs.

Normal toric ideals of low codimension

Every normal toric ideal of codimension two is minimally generated by a Gröbner basis with squarefree initial monomials. A polynomial time algorithm is presented for checking whether a toric ideal of fixed codimension is normal.

Gröbner Bases of Simplicial Toric Ideals

Bounds for the maximum degree of a minimal Gröbner basis of simplicial toric ideals with respect to the reverse lexicographic order are given. These bounds are close to the bound stated in Eisenbud-Goto’s Conjecture on the Castelnuovo-Mumford regularity.

Two way subtable sum problems and quadratic Gröbner bases

Hara, Takemura and Yoshida discussed toric ideals arising from two way subtable sum problems and showed that these toric ideals are generated by quadratic binomials if and only if the subtables are either diagonal or triangular. In the present paper, we show that if the subtables are either diagonal or triangular, then their toric ideals possess quadratic Gröbner bases.

Monomial and toric ideals associated to Ferrers graphs

Each partition A = (λ 1 , λ 2 ,..., An) determines a so-called Ferrers tableau or, equivalently, a Ferrers bipartite graph. Its edge ideal, dubbed a Ferrers ideal, is a squarefree monomial ideal that is generated by quadrics. We show that such an ideal has a 2-linear minimal free resolution; i.e. it defines a small subscheme. In fact, we prove that this property characterizes Ferrers graphs among bipartite graphs. Furthermore, using a method of Bayer and Sturmfels, we provide an explicit description of the maps in its minimal free resolution. This is obtained by associating a suitable polyhedral cell complex to the ideal/graph. Along the way, we also determine the irredundant primary decomposition of any Ferrers ideal. We conclude our analysis by studying several features of toric rings of Ferrers graphs. In particular we recover/establish formulae for the Hilbert series, the Castelnuovo-Mumford regularity, and the multiplicity of these rings. While most of the previous works in this highly investigated area of research involve path counting arguments, we offer here a new and self-contained approach based on results from Gorenstein liaison theory.

Toric geometry of cuts and splits

Associated to any graph is a toric ideal whose generators record relations among the cuts of the graph. We study these ideals and the geometry of the corresponding toric varieties. Our theorems and conjectures relate the combinatorial structure of the graph and the corresponding cut polytope to algebraic properties of the ideal. Cut ideals generalize toric ideals arising in phylogenetics and the study of contingency tables.