Binomial Ideals Research Articles

Binomial ideals in quantum tori and quantum affine spaces

The article targets binomial ideals in quantum tori and quantum affine spaces. First, noncommutative analogs of known results for commutative (Laurent) polynomial rings are obtained, including the following: Under the assumption of an algebraically closed base field, it is proved that primitive ideals are binomial, as are radicals of binomial ideals and prime ideals minimal over binomial ideals. In the case of a quantum torus Tq, the results are strongest: In this situation, the binomial ideals are parametrized by characters on sublattices of the free abelian group whose group algebra is the center of Tq; the sublattice-character pairs corresponding to primitive ideals as well as to radicals and minimal primes of binomial ideals are determined. As for occurrences of binomial ideals in quantum algebras: It is shown that cocycle-twisted group algebras of finitely generated abelian groups are quotients of quantum tori modulo binomial ideals. Another appearance is as follows: Cocycle-twisted semigroup algebras of finitely generated commutative monoids, as well as quantum affine toric varieties, are quotients of quantum affine spaces modulo certain types of binomial ideals.

Binomial ideals attached to finite collections of cells

We consider the ideal of inner 2-minors I P of a finite set of cells P , which we call the cell ideal of P . A nice interpretation for the height of an unmixed ideal I P , in terms of the number of cells of P is given. Moreover, the coordinate rings of cell ideals with isolated singularities are determined.

On binomial complete intersections

We consider homogeneous binomial ideals I=(f1,…,fn) in K[x1,…,xn], where fi=aixidi−bimi and ai≠0. When such an ideal is a complete intersection, we show that the monomials which are not divisible by xidi for i=1,…,n form a vector space basis for the corresponding quotient, and we describe the Macaulay dual generator in terms of a directed graph that we associate to I. These two properties can be seen as a natural generalization of well-known properties for monomial complete intersections. Moreover, we give a description of the radical of the resultant of I in terms of the directed graph.

Polyocollection ideals and primary decomposition of polyomino ideals

In this article, we study the primary decomposition of some binomial ideals. In particular, we introduce the concept of polyocollection, a combinatorial object that generalizes the definitions of collection of cells and polyomino, that can be used to compute a primary decomposition of non-prime polyomino ideals. Furthermore, we give a description of the minimal primary decomposition of non-prime closed path polyominoes. In particular, for such a class of polyominoes, we characterize the set of all zig-zag walks and show that the minimal prime ideals have a very nice combinatorial description.

Hadamard products and binomial ideals

We study the Hadamard product of two varieties V and W, with particular attention to the situation when one or both of V and W is a binomial variety. The main result of this paper shows that when V and W are both binomial varieties, and the binomials that define V and W have the same binomial exponents, then the defining equations of V⋆W can be computed explicitly and directly from the defining equations of V and W. This result recovers known results about Hadamard products of binomial hypersurfaces and toric varieties. Moreover, as an application of our main result, we describe a relationship between the Hadamard product of the toric ideal IG of a graph G and the toric ideal IH of a subgraph H of G. We also derive results about algebraic invariants of Hadamard products: assuming V and W are binomial with the same exponents, we show that deg⁡(V⋆W)=deg⁡(V)=deg⁡(W) and dim⁡(V⋆W)=dim⁡(V)=dim⁡(W). Finally, given any (not necessarily binomial) projective variety V and a point p∈Pn∖V(x0x1⋯xn), subject to some additional minor hypotheses, we find an explicit binomial variety that describes all the points q that satisfy p⋆V=q⋆V.

Derivation modules for sum and gluing

In this paper, we explicitly compute the derivation module of quotients of polynomial rings by ideals formed by the sum or by some other gluing technique. We discuss cases of monomial ideals and binomial ideals separately.

A determinantal formula for circuits of integer lattices

Abstract Let L be a not necessarily saturated lattice in ℤ n with a defining matrix B. We explicitly compute the set of circuits of L in terms of maximal minors of B. This has a variety of applications from toric to tropical geometry, from Gröbner to Graver bases, and from linear to binomial ideals.

CRITICAL BINOMIAL IDEALS OF NORTHCOTT TYPE

AbstractIn this paper, we study a family of binomial ideals defining monomial curves in the n-dimensional affine space determined by n hypersurfaces of the form $x_i^{c_i} - x_1^{u_{i1}} \cdots x_n^{u_{1n}}$ in $\Bbbk [x_1, \ldots , x_n]$ with $u_{ii} = 0, \ i\in \{ 1, \ldots , n\}$ . We prove that the monomial curves in that family are set-theoretic complete intersections. Moreover, if the monomial curve is irreducible, we compute some invariants such as genus, type and Frobenius number of the corresponding numerical semigroup. We also describe a method to produce set-theoretic complete intersection semigroup ideals of arbitrary large height.

Regularity of powers of quadratic sequences with applications to binomial ideals

In this article, we obtain an upper bound for the Castelnuovo-Mumford regularity of powers of an ideal generated by a homogeneous quadratic sequence in a polynomial ring in terms of the regularity of its related ideals and degrees of its generators. As a consequence, we compute upper bounds for the regularity of powers of several binomial ideals. We generalize a result of Matsuda and Murai to show that the regularity of JGs is bounded below by 2s+ℓ(G)−1 for all s≥1, where JG denotes the binomial edge ideal of a graph G and ℓ(G) is the length of a longest induced path in G. We compute the regularity of powers of binomial edge ideals of cycle graphs, star graphs, and balloon graphs explicitly. Also, we give sharp bounds for the regularity of powers of almost complete intersection binomial edge ideals and parity binomial edge ideals.

Toric degenerations of Grassmannians and Schubert varieties from matching field tableaux

We study Gröbner degenerations of Grassmannians and the Schubert varieties inside them. We provide a family of binomial ideals whose combinatorics is governed by matching field tableaux in the sense of Sturmfels and Zelevinsky in [31]. We prove that these ideals are all quadratically generated and they yield a SAGBI basis of the Plücker algebra. This leads to a new family of toric degenerations of Grassmannians. Moreover, we apply our results to construct a family of Gröbner degenerations of Schubert varieties inside Grassmannians. We provide a complete characterization of toric ideals among these degenerations in terms of the combinatorics of matching fields, permutations and semi-standard tableaux.

Corrigendum: Decompositions of cellular binomial ideals

This note contains a correct proof of Theorem 2.11 from the article Decompositions of cellular binomial ideals.

Polynomial expressions for non-binomial structures

Following recent paper of the author about polynomial expressions with respect to binomial ideals, the current paper is devoted to the non-binomial case. Beside the established facts in the underlying theory, the correctness and termination of the algorithms are addressed. From the collection of examples, the main large-scale non-binomial one is of particular importance. Application of the new algorithms for the cases of moderately large binomial like cyclic 16-roots and cyclic 18-roots is discussed. A combinatorial analysis for complexity of the proposed algorithms is shown to be at least O(log⁡(log⁡n)n2(log⁡n)2).

Some algebraic aspects of mesoprimary decomposition

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Monoid congruences (and therefore, binomial ideals) can present many subtle behaviors that must be carefully accounted for in order to produce general results, and this makes the theory complicated. In this paper, we examine their results in the presence of a positive A-grading, where certain pathologies are avoided and the theory becomes more accessible. Our approach is algebraic: while key notions for mesoprimary decomposition are developed first from a combinatorial point of view, here we state definitions and results in algebraic terms, which are moreover significantly simplified due to our (slightly) restricted setting. In the case of toral components (which are well-behaved with respect to the A-grading), we are able to obtain further simplifications under additional assumptions. We also provide counterexamples to two open questions, identifying (i) a binomial ideal whose hull is not binomial, answering a question of Eisenbud and Sturmfels, and (ii) a binomial ideal I for which Itoral is not binomial, answering a question of Dickenstein, Miller and the first author.

Binomial edge ideals of bipartite graphs

Binomial edge ideals are a noteworthy class of binomial ideals that can be associated with graphs, generalizing the ideals of 2-minors. For bipartite graphs we prove the converse of Hartshorne’s Connectedness Theorem, according to which if an ideal is Cohen–Macaulay, then its dual graph is connected. This allows us to classify Cohen–Macaulay binomial edge ideals of bipartite graphs, giving an explicit and recursive construction in graph-theoretical terms. This result represents a binomial analogue of the celebrated characterization of (monomial) edge ideals of bipartite graphs due to Herzog and Hibi (2005).

Mesoprimary decomposition of binomial submodules

Recent results of Kahle and Miller give a method of constructing primary decompositions of binomial ideals by first constructing “mesoprimary decompositions” determined by their underlying monoid congruences. Mesoprimary decompositions are highly combinatorial in nature, and are designed to parallel standard primary decomposition over Noetherean rings. In this paper, we generalize mesoprimary decomposition from binomial ideals to “binomial submodules” of certain graded modules over a monoid algebra, analogous to the way primary decomposition of ideals over a Noetherean ring R generalizes to R-modules. The result is a combinatorial method of constructing primary decompositions that, when restricting to the special case of binomial ideals, coincides with the method introduced by Kahle and Miller.

Betti numbers of binomial ideals

Let us consider the family of binomial ideals B=I+J, where J is lattice ideal and I is a square-free quadratic monomial ideal. We give a formula for calculating the Betti numbers of B. Moreover we bound the Green–Lazarsfeld invariant of a family of quadratic binomial ideals B using this formula. This result extends a previous result of Eisenbud et al. for square-free quadratic monomial ideals and extends completely Fröberg's theorem. We describe also a subfamily where we can calculate the Green–Lazarsfeld invariant of any ideal B and we also compute its first non-linear Betti number.

Irreducible decomposition of binomial ideals

Building on coprincipal mesoprimary decomposition [Kahle and Miller, Decompositions of commutative monoid congruences and binomial ideals, Algebra and Number Theory 8 (2014), 1297–1364], we combinatorially construct an irreducible decomposition of any given binomial ideal. In a parallel manner, for congruences in commutative monoids we construct decompositions that are direct combinatorial analogues of binomial irreducible decompositions, and for binomial ideals we construct decompositions into ideals that are as irreducible as possible while remaining binomial. We provide an example of a binomial ideal that is not an intersection of irreducible binomial ideals, thus answering a question of Eisenbud and Sturmfels [Binomial ideals, Duke Math. J. 84 (1996), 1–45].

Detecting binomiality

Binomial ideals are special polynomial ideals with many algorithmically and theoretically nice properties. We discuss the problem of deciding if a given polynomial ideal is binomial. While the methods are general, our main motivation and source of examples is the simplification of steady state equations of chemical reaction networks. For homogeneous ideals we give an efficient, Gröbner-free algorithm for binomiality detection, based on linear algebra only. On inhomogeneous input the algorithm can only give a sufficient condition for binomiality. As a remedy we construct a heuristic toolbox that can lead to simplifications even if the given ideal is not binomial.

On Monomial Reduction and Polynomial Expressions with Respect to Binomial Ideals

Let I ⊂ K[x 1,…, x n ] be an ideal and G be the reduced Gröbner basis of I with respect to lexicographic monomial order. We introduce the index of an expression of f ∈ K[x 1,…, x n ] with respect to G. A minimal expression is characterized as the one with zero G-index. In case where I is a binomial prime ideal, a new division algorithm with minimal and unique expression is presented. The application of our new method on benchmark polynomial systems cyclic-9 and cyclic-12 shows its superiority in comparison with the existing division algorithm.

Binomial ideal associated to a lattice and its label code

In coding theory the study of the binomial ideal derived from an arbitrary code is currently of great interest; see for example [5]. This is mainly because of a known relation between binomial ideals and lattices or codes. Also, studying the relation between binomial ideals associated to a lattice and its label code helps to solve the closest vector problem in lattices as well as decoding binary and non-binary codes [1, 3] and finding a label code of a lattice, as we do in this work. Every lattice Λ can be described in terms of a label code L and an orthogonal sublattice Λ′ such that Λ/Λ′ ∼= L [2]. We assign binomial ideals IΛ and IL to an integer lattice Λ and its label code L, respectively. In this work, we identify the binomial ideal associated to an integer lattice and then establish the relation IΛ = IΛ′ + IL between the ideal of the lattice and its label code. In this work, we define a binomial ideal for an integer lattice and its label code slightly different from [1, 3, 4, 7]. Let K[X] = K[x1, . . . , xn] denote the polynomial ring, where K is an arbitrary field. Consider ≺ as a fixed total degree compatible term order with x1 x2 · · · xn. The monomials in K[X] are denoted by X = x1 1 . . . x bn n where b = (b1, . . . , bn) is an element of N0 and N0 is the set of non-negative integers. We use the notation