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

Let $A={{bf a}_1,...,{bf a}_m} subset mathbb{Z}^n$ be a vector configuration and $I_A subset K[x_1,...,x_m]$ 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 $Delta _{ind(A)}$. We show that the vertices of $Delta_{ind(A)}$ correspond to the indispensable monomials of the toric ideal $I_A$, while one dimensional facets of $Delta_{ind(A)}$ with minimal binomial $A$-degree correspond to the indispensable binomials of $I_{A}$.