Binomial fibers and indispensable binomials

Abstract

Abstract. Let I be an arbitrary ideal generated by binomials. We show that certain equivalence classes of fibers are associated to any minimal binomial generating set of I. We provide a simple and efficient algorithm to compute the indispensable binomials of a binomial ideal from a given generating set of binomials and an algorithm to detect whether a binomial ideal is generated by indispensable binomials.

Minimal generating sets of lattice ideals

“…Note that in the case of positive lattices the notions of indispensable and weakly indispensable coincide. Recently, a polynomial-time algorithm for computing all indispensable binomials from a given system of binomial generators of an arbitrary binomial ideal was given in [7,Algorithm 1]. Next we complete the classification of indispensable binomials/monomials of lattice ideals in the case of non-positive lattices.…”

Minimal set of binomial generators for certain Veronese 3-fold projections

“…, a t } ⊂ Z n+1 + satisfying a 1 = α + , a t = α − and a i − a i+1 ∈ D for all 1 ≤ i ≤ t. In [9], Diaconis and Sturmfels showed that given a set of generators D of L then I(D) = I L if and only if D is a Markov basis. We cite [5], [6], [7], [9] and [14] for a detailed exposition of Markov bases of lattice ideals and related problems.…”

The Universal Gröbner Basis of a Binomial Edge Ideal

“…Graver bases were originally defined for toric ideals by Sturmfels [Stu96]. Charalambous, Thoma and Vladoiu [CTV16] recently generalised the concept to an arbitrary pure difference ideal I, showing in particular that Gr(I) is finite and includes U(I) as a subset.…”