Mixed matrices and binomial ideals

Affine semigroup rings that are complete intersections

“…This statement was proved for semigroups of dimension one by C. Delorme [D] and by [FS2], and recently by J.C. Rosales and P.A. García-Sánchez [RG-S] for semigroups with dimension less than 4.…”

“…We will call a matrix M mixed if every row of M contains a positive and a negative entry. We will call M dominating if it contains no square mixed submatrix (see [FS2], Proposition 2.6, where the name is motivated).…”

Set-theoretic complete intersections on binomials

“…Now we prove that M is dominating, i.e., no square submatrix of M is mixed. The following proof is inspired by 2.3 of [7]. Assume that N is a mixed s × s submatrix of M , with s ≥ 1 and suppose that s is maximal with respect to this property.…”

“…Eisenbud and Sturmfels began the systematic study of binomial ideals in [5], where the ubiquity of binomial ideals was also presented. There have been numerous publications in recent years on binomial ideals, and several of them treat the problem of the minimal generation of a binomial ideal or of the radical of it; for example, [1,3,6,7,8,10,12,13,15].…”

Complete intersections in toric ideals

“…It is from this perspective, and beginning with the work of Herzog [13] and Delorme [5], that the question of classifying complete intersection binomial ideals has been extensively studied by many authors [1,2,3,9,10,11,14,15,16,17,18,20,21,22]. A combinatorial characterization of these ideals is given in [11] in terms of a choice of B and the notion of mixed matrices; that is, matrices such that every column contains a strictly positive and a strictly negative entry. Note that since the columns of the matrix B add up to zero, B is automatically mixed.…”