Universal Grobner Basis Research Articles

On the Grobner Basis of the Toric Ideal for 3 X n- Contingency Tables

In this paper, The Grobner basis of the Toric Ideal for - contingency tables related with the Markov basis B introduced by Hussein S. MH, Abdulrahman H. M in 2018 is found. Also, the Grobner basis is a reduced and universal Grobner basis are shown.

Vanishing Ideals of Projective Spaces over Finite Fields and a Projective Footprint Bound

We consider the vanishing ideal of a projective space over a finite field. An explicit set of generators for this ideal has been given by Mercier and Rolland. We show that these generators form a universal Grobner basis of the ideal. Further we give a projective analogue for the so-called footprint bound, and a version of it that is suitable for estimating the number of rational points of projective algebraic varieties over finite fields. An application to Serre’s inequality for the number of points of projective hypersurfaces over finite fields is included.

Toric ideals and diagonal 2-minors

Let G be a simple graph on the vertex set \({\{1,\ldots,n\}}\) with m edges. An algebraic object attached to G is the ideal PG generated by diagonal 2-minors of an \({n \times n}\) matrix of variables. We prove that if G is bipartite, then every initial ideal of PG is generated by squarefree monomials of degree at most \({\lfloor{\frac{m+n+1}{2}} \rfloor}\). Furthermore, we completely characterize all connected graphs G for which PG is the toric ideal associated to a finite simple graph. Finally we compute in certain cases the universal Grobner basis of PG.

Robust Graph Ideals

Let I be a toric ideal. We say I is robust if its universal Grobner 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 characterize all graphs which give rise to robust ideals. Our characterization shows that robustness can be determined solely in terms of graph-theoretic conditions on the set of circuits of G.

The closure of a linear space in a product of lines

Given a linear space L in affine space $${\mathbb {A}}^n$$An, we study its closure $$\widetilde{L}$$L~ in the product of projective lines $$({\mathbb {P}}^1)^n$$(P1)n. We show that the degree, multigraded Betti numbers, defining equations, and universal Grobner basis of its defining ideal $$I(\widetilde{L})$$I(L~) are all combinatorially determined by the matroid M of L. We also prove that $$I(\widetilde{L})$$I(L~) and all of its initial ideals are Cohen---Macaulay with the same Betti numbers, and can be used to compute the h-vector of M. This variety $$\widetilde{L}$$L~ also gives rise to two new objects with interesting properties: the cocircuit polytope and the external activity complex of a matroid.

Universal Gröbner basis associated with the maximum flow problem

We give a formulation of the maximum flow problem as an integer programming problem in the standard form. We characterize elementary vectors of the kernel lattice of the matrix coefficient in our formulation in terms of the combinatorial property of the graph, such as circuits and s–t paths, and we determine the universal Grobner basis of the toric ideal associated with the maximum flow problem. Next, we examine the kernel lattice of the reduced incidence matrix of the digraph. Under suitable assumptions for the digraph, we prove that it is generated by all incidence vectors of s–t directed paths, which yield the generator of the toric ideal associated with the reduced incidence matrix of the digraph.

An Algebraic Exploration of Dominating Sets and Vizing's Conjecture

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two different ways: first, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a sufficient criterion for demonstrating that a particular ideal representation is already the universal Grobner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Grobner basis for that ideal. We also present the first algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational ramifications to this conjecture when using either of the two dominating set representations described above.

Free resolutions and sparse determinantal ideals

A sparse generic matrix is a matrix whose entries are distinct variables and zeros. Such matrices were studied by Giusti and Merle who computed some invariants of their ideals of maximal minors. In this paper we extend these results by computing a minimal free resolution for all such sparse determinantal ideals. We do so by introducing a technique for pruning minimal free resolutions when a subset of the variables is set to zero. Our technique correctly computes a minimal free resolution in two cases of interest: resolutions of monomial ideals, and ideals resolved by the Eagon-Northcott Complex. As a consequence we can show that sparse determinantal ideals have a linear resolution over the integers, and that the projective dimension depends only on the number of columns of the matrix which are identically zero. Finally, we show that all such ideals have the property that regardless of the term order chosen, the Betti numbers of the ideal and its initial ideal are the same. In particular the nonzero generators of these ideals form a universal Grobner basis.

Universal and comprehensive Gröbner bases of the classical determinantal ideal

Let A =(x ij ), i =1,2,... ,k, j =1,2,... ,l, 1 ≤ k ≤ l, be a matrix of independent variables, G be the set of maximal minors of A, and I = (G) be the classical determinantal ideal. We show that G is a universal Grobner basis of I. Also, a sufficient condition for G to be a universal comprehensive Grobner basis is proved. Bibliography: 12 titles.

Enumeration of finite monomial orderings and combinatorics of universal Gröbner bases

The goal of this work is to analyze various classes of finite and total monomial orderings. The concept of monomial ordering plays the key role in the theory of Grobner bases: every basis is determined by a certain ordering. At the same time, in order to define a Grobner basis, it is not necessary to know ordering of all monomials. Instead, it is sufficient to know only a finite interval of the given ordering. We consider combinatorics of finite monomial orderings and its relationship with cells of a universal Grobner basis. For every considered class of orderings (weakly admissible, convex, and admissible), an algorithm for enumerating finite orderings is discussed and combinatorial integer sequences are obtained. An algorithm for computing all minimal finite orderings for an arbitrary Grobner basis that completely determine this basis is presented. The paper presents also an algorithm for computing an extended universal Grobner basis for an arbitrary zero-dimensional ideal.

Variation of cost functions in integer programming

We study the problem of minimizingc · x subject toA · x =b. x ź 0 andx integral, for a fixed matrixA. Two cost functionsc andcź are considered equivalent if they give the same optimal solutions for eachb. We construct a polytopeSt(A) whose normal cones are the equivalence classes. Explicit inequality presentations of these cones are given by the reduced Grobner bases associated withA. The union of the reduced Grobner bases asc varies (called the universal Grobner basis) consists precisely of the edge directions ofSt(A). We present geometric algorithms for computingSt(A), the Graver basis, and the universal Grobner basis.

Gröbner bases and triangulations of the second hypersimplex

The algebraic technique of Grobner bases is applied to study triangulations of the second hypersimplex Δ(2,n). We present a quadratic Grobner basis for the associated toric idealK(K n ). The simplices in the resulting triangulation of Δ(2,n) have unit volume, and they are indexed by subgraphs which are linear thrackles [28] with respect to a circular embedding ofK n . Forn≥6 the number of distinct initial ideals ofI(K n ) exceeds the number of regular triangulations of Δ(2,n); more precisely, the secondary polytope of Δ(2,n) equals the state polytope ofI(K n ) forn≤5 but not forn≥6. We also construct a non-regular triangulation of Δ(2,n) forn≥9. We determine an explicit universal Grobner basis ofI(K n ) forn≤8. Potential applications in combinatorial optimization and random generation of graphs are indicated.

Combinatorics of Maximal Minors

We continue the study of the Newton polytope ∏m,n of the product of all maximal minors of an m × n-matrix of indeterminates. The vertices of ∏m,n are encoded by coherent matching fields Λ e (Λσ), where σ runs over all m-element subsets of columns, and each Λσ is a bijection σ → lmr. We show that coherent matching fields satisfy some axioms analogous to the basis exchange axiom in the matroid theory. Their analysis implies that maximal minors form a universal Grobner basis for the ideal generated by them in the polynomial ring. We study also another way of encoding vertices of ∏m,n for m ≤ n by means of “generalized permutations”, which are bijections between (n − m + 1)–element subsets of columns and (n − m + 1)–element submultisets of rows.

ASYMPTOTIC ANALYSIS OF TORIC IDEALS

This partially expository note extends and refines our earlier work on Grobner bases of toric varieties [14]. It is closely related to the theory of A-hypergeometric functions due to Gel′fand Graev, Kapranov and Zelevinsky (see § 6). The main new result in this note is the characterization of the universal Grobner basis of a unimodular toric ideal (see § 5).