Minimal systems of binomial generators and the indispensable complex of a toric ideal

Abstract

Abstract. Let A = {a 1 , . . . , am} ⊂ Z n be a vector configuration and I A ⊂ K[x 1 , . . . , xm] its corresponding toric ideal. The paper consists of two parts. In the first part we completely determine the number of different minimal systems of binomial generators of I A . We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to A a simplicial complex ∆ ind(A) . We show that the vertices of ∆ ind(A) correspond to the indispensable monomials of the toric ideal I A , while one dimensional facets of ∆ ind(A) with minimal binomial A-degree correspond to the indispensable binomials of I A .

Arithmetical rank of toric ideals associated to graphs

“…Consequently there are at least two indispensable binomials with the same G-degree, contradicting Theorem 3.4 in [2]. Recall that the G-degree of a binomial …”

“…From Proposition 3.1 in [2] we have that the set of indispensable monomials is the unique minimal generating set of N G . The following lemma will be useful in the proof of Theorem 3.2 and Proposition 3.4.…”

Binomial fibers and indispensable binomials

“…Up to now, it has mainly been addressed for toric and lattice ideals, see [3,4,5,11,12,22] among others. In Section 1, we consider this problem in the case of binomial ideals.…”

“…When I := I L is the lattice ideal of L, the equivalence class of u consists precisely of all v such that u − v ∈ L and the I-fibers are finite exactly when L ∩ N n = {0}. In this case, for each I-fiber one can use a graph construction, see [7,4], that determines the I-fibers that appear as invariants associated to any minimal generating set of I. We also note that in [5] the fibers of I L were studied even when L ∩ N n = {0}.…”

“…In [4, Proposition 3.1], it was shown that to find the indispensable monomials of I L , it is enough to consider any one of the binomial generating sets of I L . Moreover in [4,Theorem 2.12] it was shown that in order to find the indispensable binomials of I L , it is enough to consider any minimal binomial generating set of I L , assign Z n /L-degrees to the binomials of this set and to compute their minimal Z n /L-degrees. More recently in [14, Theorem 1.1, Corollary 1.3], it was shown that if I is a pure binomial ideal then there is a d ∈ N such that any I is A-graded for some A ⊂ Z d : when NA ∩ (−NA) = {0} and all fibers are finite a sufficient condition was given in [14] for the indispensable binomials and a characterization for the indispensable monomials, involving the I-fibers of a minimal generating set of I.…”