Toric Ideals Research Articles

Comparability of the total Betti numbers of toric ideals of graphs

Comparability of the total Betti numbers of toric ideals of graphs

Universally free numerical semigroups

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 generated by their set of circuits. Additionally, we provide a characterization of universally free numerical semigroups in terms of Gröbner bases. Specifically, a numerical semigroup is universally free if and only if all initial ideals of its corresponding toric ideal are complete intersections. Furthermore, we establish several equalities among the toric bases of a universally free numerical semigroup.We provide a complete characterization of 3-generated universally free numerical semigroups in terms of their minimal generating sets, and by proving the equality of certain toric bases. We compute exactly all the toric bases of a toric ideal defined by a 3-generated universally free numerical semigroup. Notably, we answer some questions posed by Tatakis and Thoma by demonstrating that toric ideals defined by 3-generated universally free numerical semigroups have a set of circuits and a universal Gröbner basis of size 3, while the universal Markov basis and the Graver basis can be arbitrarily large. We present partial results and propose several conjectures regarding universally free numerical semigroups with more than three generators.

Geometric vertex decomposition and liaison for toric ideals of graphs

Geometric vertex decomposition and liaison for toric ideals of graphs

Comparing invariants of toric ideals of bipartite graphs

Let G G be a finite simple graph and let I G I_G denote its associated toric ideal in the polynomial ring R R . For each integer n ≥ 2 n\geq 2 , we completely determine all the possible values for the tuple ( reg ⁡ ( R / I G ) , deg ⁡ ( h R / I G ( t ) ) , pdim ⁡ ( R / I G ) , depth ⁡ ( R / I G ) , dim ⁡ ( R / I G ) ) (\operatorname {reg}(R/I_G), \deg (h_{R/I_G}(t)), \operatorname {pdim}(R/I_G), \operatorname {depth}(R/I_G), \dim (R/I_G)) when G G is a connected bipartite graph on n n vertices.

On the Hilbert series of the tangent cones for some 4-generated pseudosymmetric monomial curves

In this article, we study Hilbert Series of non-Cohen-Maculay tangent cones for some 4-generated pseudo symmetric monomial curves. We show that the Hilbert Function is nondecreasing by explicitly computing it. We also compute standard bases of these toric ideals.

The Markov complexity of book graphs

Let A be the incidence matrix of a book graph with expandable pages, A(r) its r-th Lawrence lifting and IA(r) the corresponding toric ideal, where r≥3. We compute the minimal systems of generators for the double family of toric ideals IA(r). We prove that toric ideals of book graphs with expandable pages may have arbitrary large Markov complexity, by explicitly computing it.

On the strongly robust property of toric ideals

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 and non-mixed. We prove that the cardinality of the following sets - the set of indispensable elements, minimal Markov bases, the Universal Markov basis and the Universal Gröbner basis of a toric ideal - depends only on the type of the bouquets and the bouquet ideal. These results enable us to introduce the strongly robust simplicial complex and show that it determines the strongly robust property. For codimension 2 toric ideals, we study the strongly robust simplicial complex and prove that robust implies strongly robust.

Oriented graphs whose toric ideals are a complete intersection

Oriented graphs whose toric ideals are a complete intersection

Low dimensional flow polytopes and their toric ideals

The toric ideal of a d-dimensional flow polytope has an initial ideal generated by square-free monomials of degree at most d. The toric ideal of a flow polytope of dimension at most four has an initial ideal generated by square-free monomials of degree at most two, with the only exception of the four-dimensional Birkhoff polytope, whose toric ideal has an initial ideal generated by a square-free cubic monomial. The proof is based on a method to classify certain compressed flow polytopes, and a construction of a quadratic pulling triangulation of them. Along the way compressed flow polytopes are classified up to dimension four, and their Ehrhart polynomials are computed.

State polytopes related to two classes of combinatorial neural codes

Combinatorial neural codes are collections of 0/1 vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn on the plane as a Venn diagram-like figure. A sufficient condition to do so is for the code to have a property called 2-inductively pierced. Gross, Obatake, and Youngs recently used toric algebra to show that a code on three neurons is 1-inductively pierced if and only if the toric ideal is trivial or generated by quadratics. No result is known for additional neurons in the same generality, part of the difficulty coming from the large number of codewords possible when additional neurons are used.In this article, we study two infinite classes of combinatorial neural codes in detail. For each code, we explicitly compute its universal Gröbner basis. This is done for the first class by recognizing that the codewords form a Lawrence-type matrix. With the second class, this is done by showing that the matrix is totally unimodular. These computations allow one to compute the state polytopes of the corresponding toric ideals, from which all distinct initial ideals may be computed efficiently. Moreover, we show that the state polytopes are combinatorially equivalent to well-known polytopes: the permutohedron and the stellohedron.

On robustness and related properties on toric ideals

A toric ideal is called robust if its universal Gröbner basis is a minimal set of generators, and is called generalized robust if its universal Gröbner 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 by-product, 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 Gröbner 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.

Directed Gaussian graphical models with toric vanishing ideals

Directed Gaussian graphical models are statistical models that use a directed acyclic graph (DAG) to represent the conditional independence structures between a set of jointly normal random variables. The DAG specifies the model through recursive factorization of the parametrization, via restricted conditional distributions. In this paper, we make an attempt to characterize the DAGs whose vanishing ideals are toric ideals. In particular, we give some combinatorial criteria to construct such DAGs from smaller DAGs which have toric vanishing ideals. An associated monomial map called the shortest trek map plays an important role in our description of toric Gaussian DAG models. For DAGs whose vanishing ideal is toric, we prove results about the generating sets of those toric ideals.

Toric ideals of weighted oriented graphs

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 depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, unoriented graph case, to show that when the associated simple graph has only trivial even closed walks, the toric ideal is the zero ideal. Moreover, we give necessary and sufficient conditions for the toric ideal of a weighted oriented graph to be generated by a single binomial and we describe the binomial in terms of the structure of the graph.

Quadratic Gröbner bases of block diagonal matching field ideals and toric degenerations of Grassmannians

In the present paper, we prove that the toric ideals of certain s -block diagonal matching fields have quadratic Gröbner bases. Thus, in particular, those are quadratically generated. By using this result, we provide a new family of toric degenerations of Grassmannians.

Gröbner Bases of Toric Ideals Associated with Matroids

In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange.

Universal Gröbner bases of toric ideals of combinatorial neural codes

In the 1970s, O'Keefe and Dostrovsky discovered that certain neurons, called place cells, in an animal's brain are tied to its location within its arena. A combinatorial neural code is a collection of $0/1$-vectors which encode the patterns of co-firing activity among the place cells. Gross, Obatake, and Youngs have recently used techniques from toric algebra to study when a neural code is $0$- $1$-, or $2$-inductively pierced: a property that allows one to reconstruct a Venn diagram-like planar figure that acts as a geometric schematic for the neural co-firing patterns. This article examines their work closely by focusing on a variety of classes of combinatorial neural codes. In particular, we identify universal Gr\"obner bases of the toric ideal for these codes.

Hierarchical Aitchison–Silvey models for incomplete binary sample spaces

Multivariate sample spaces may be incomplete Cartesian products, when certain combinations of the categories of the variables are not possible. Traditional log-linear models, which generalize independence and conditional independence, do not apply in such cases, as they may associate positive probabilities with the non-existing cells. To describe the association structure in incomplete sample spaces, this paper develops a class of hierarchical multiplicative models which are defined by setting certain non-homogeneous generalized odds ratios equal to one and are named after Aitchison and Silvey who were among the first to consider such ratios. These models are curved exponential families that do not contain an overall effect and, from an algebraic perspective, are non-homogeneous toric ideals. The relationship of this model class with log-linear models and quasi log-linear models is studied in detail in terms of both statistics and algebraic geometry. The existence of maximum likelihood estimates and their properties, as well as the relevant algorithms are also discussed.

Binomial ideals of domino tilings

In this paper, we consider the set of all domino tilings of a cubiculated region. The primary question we explore is: How can we move from one tiling to another? Tiling spaces can be viewed as spaces of subgraphs of a fixed graph with a fixed degree sequence. Moves to connect such spaces have been explored in algebraic statistics. Thus, we approach this question from an applied algebra viewpoint, making new connections between domino tilings, algebraic statistics, and toric algebra. Using results from toric ideals of graphs, we are able to describe moves that connect the tiling space of a given cubiculated region of any dimension. This is done by studying binomials that arise from two distinct domino tilings of the same region. Additionally, we introduce tiling ideals and flip ideals and use these ideals to restate what it means for a tiling space to be flip connected. Finally, we show that if R is a 2-dimensional simply connected cubiculated region, any binomial arising from two distinct tilings of R can be written in terms of quadratic binomials. As a corollary to our main result, we obtain an alternative proof to the fact that the set of domino tilings of a 2-dimensional simply connected region is connected by flips.

Equivariant Hilbert series for hierarchical Models

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We present a condition that guarantees that the equivariant Hilbert series is a rational function in m+1 variables with rational coefficients. Furthermore we give explicit formulas for the rational functions with coefficients in a number field and an algorithm for determining the rational functions with rational coefficients. A key is to construct finite automata that recognize languages corresponding to invariant filtrations.

On the toric ideals of matroids of a fixed rank

In 1980 White conjectured that every element of the toric ideal of a matroid is generated by quadratic binomials corresponding to symmetric exchanges. We prove White’s conjecture for high degrees with respect to the rank. This extends our result (Lasoń and Michałek in Adv Math 259:1–12, 2014) confirming White’s conjecture ‘up to saturation’. Furthermore, we study degrees of Gröbner bases and Betti tables of the toric ideals of matroids of a fixed rank.