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

Let A = {a1, . . . , am} � Zn be a vector configuration and IA � K(x1, . . . , 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 IA. We also prove that generic toric ideals are generated by indispensable binomials. In the second part we associate to A a simplicial complexind(A). We show that the vertices ofind(A) correspond to the indispensable monomials of the toric ideal IA, while one dimensional facets ofind(A) with minimal binomial A-degree correspond to the indispensable binomials of IA.